There's nothing wrong with wanting to put yourself in the story and writing about wonderful and beautiful things happening. Nothing at all. It's a great deal of fun for you; if you're writing your friends in it, it's a great deal of fun for them as well. It isn't likely to be a universal classic, though -- unless everyone knows you and your friends, and likes you and your friends, they're probably not going to be interested enough to read it, and would probably prefer to avoid it if given the option. It is possible that you've written a universal classic, but the odds are very much against it.
Whatever you do, don't stop writing. All this writing that you're doing is helping you hone your technical writing craft, even though there will be places that very much need some work. Idealistic self-insert stories may not help you learn some aspects of plotting or modern three-dimensional characterization, but they're definitely practice for description and narrative. A year later, you may look back at what you wrote and immediately see thirty flaws on every page. Realize that your ability to see those flaws is because of all that practice and experience you've racked up. I spent ten years working on a single book. I kept writing on it, and every now and then I would go back through and realize that one section or another did not stand up to the quality of the rest of the book. So I'd go through and re-write enormous sections of it. Some sections wound up being re-written several times. The direct result of all this literary navel-gazing is my current ability to write huge chunks of very readable first draft very, very fast.
If you have scenes that have to be cut, for gods' sake save them somewhere! Give each of them their own file, or make a big file with all the cut scenes. You never know when you're going to want to use that again, and if you have some turn of phrase you really like, don't throw it out just because it doesn't fit into your vision of the moment. Likewise, don't be scared to cut out a scene if it doesn't fit with the rest of the piece, even if you really really love it. You can save a copy of it, you can turn it into a stand-alone work, or at the very least you can put it up as an outtake. I have enough material in that would-have-been novel to write at least five books. The novel would have been huge and rambling and not at all good, but if I shredded it to pieces and sifted it into topic and found a plot for each topic in the book, I'd have about five books and each of them half-done.
Consider where you're sharing this story. Given that this is no longer the Century of the Fruitbat, you probably have it up online in some fanfiction archive or other, or in your journal, and you have the summary of the story written to be aimed directly at your intended audience -- your closest friends, the ones you're writing this to share with. The trouble with this is that while the story is your private little party, and you really wouldn't mind if the general public became friends with you and shared in the fun, the general public is not likely to share in your happiness with your shiny and would-be utopic (or dark and grim and would-be dystopic) bit of fanfiction. They're expecting fanfiction shared in that much public to be fanfiction intended for sharing with a wider and less specialized audience (all Harry Potter fans who like Hermione/Harry, for example, rather than all Harry Potter fans who like Hermione/Harry and are also your friends). If someone expecting a story of wide appeal comes in and winds up mistakenly reading your story of very narrow appeal, you may wind up in possession of a stinging review. And oh, how those fuckers hurt.
Instead of sharing on a fanfiction archive where anyone looking for the pairing you like can stumble across your fic by accident, consider archiving it only in your journal. The people who matter are going to wind up there anyway, and you can always post it to your favorite fanfiction archive site later, if the response you get in your journal from people who aren't close friends of yours is good enough to suggest that your story has wider appeal. Consider labeling your story with a summary that includes "Original character who is an idealized version of me", or "How would my friends and I fit into canon?" If people who really don't want to read those sorts of stories know this up front, then they'll be more likely to avoid your story and move on to something more to their taste.
Consider what you want to convey with the story. If you want to make the story be about someone having cool and fun things happen to them, go ahead and run with the plot where the idealized version of yourself winds up in that situation and everything works out perfectly, with only minor setbacks. (Stories like this sometimes grow up to be mainstream romance novels. If you're very good at writing this kind of story, there may be a career in it for you.)
If you want to write a story about someone being faced with serious challenges and overcoming them, make sure that there is a good chance that they might not make it through their difficulties with everything being OK. If you're serious about wanting to write a story with big obstacles for your characters to overcome, write in a few obstacles that they fail at in the course of the plot. Write in some obstacles that they only partially make their way through. If your character is struggling with a big test, having her ace the test with no problems and wind up getting every question right may be deeply satisfying to the character, but does not convey struggle very well. Having the character encounter problems getting through the test, and wind up with a B conveys that despite the problems with the test, the character did very well and succeeded at overcoming the obstacle, even though that grade was worse than the grade of the character who had very little problem with the test.
A lot of idealized characters are stunningly beautiful, with perfect skin, lovely bodies, unique eye color, perfect hair in unusual colors, and so forth. If your idealized character has any of these things going on, or other things like special powers or something, consider giving some of these things (or if not those exact things, things similar to them) to those around your idealized character. If some of the other characters around your idealized character are supposed to be drab or even ugly, play that up as well. Why have a villain who looks like an ordinary man when you can have a villain who has glowing red eyes and a hint of scaliness about him? Why have a sidekick who looks ordinary when you could have a sidekick who looks so ordinary that he can never be accurately described by people who meet him because he's just that boring? Have fun with all the character descriptions, not just one or two! 100% unique characters in a world full of ordinary people do stand out, but it's not a distinction to be used lightly.
[Edit: More about appearance.]
If you're playing with characters who are people you know, but they haven't told you that they want to be in the story you're writing, insert some plausible deniability into the situation by renaming everyone. If they find the story (or if someone else finds the story and tells them) you can always claim to have been experimenting and making a composite based on several people. You can't really do that if you've got their name on their character. Similarly, if you're writing a how-things-might-have-been story based on the events of your real life, also go back through and change the names the moment the alternate universe situation sets in. You can keep a private (or not-so-private) casting sheet showing who is played by who, but consider the likely reaction of the mean biology teacher, if she should stumble upon a piece of fiction depicting a clone of her with her own name torturing puppies. It is not likely to be a good reaction. Even a fictional likeness that you think is as accurate and true to life as you can get it is likely to be disturbing to someone who doesn't want to think about the way that other people see them.
Speaking of names, try keeping your naming consistent. If everyone is named rather normal names, someone with a beautiful and exotic name may wind up appearing rather silly in contrast to everyone else, instead of achieving that beautiful exotic effect you were dreaming of. Try giving everyone lovely names, or search for a name for your character that's still lovely without appearing out of place compared to everyone else.
If you're writing this not just because you're telling a story that's fun, but because the story has a lot of deeper meaning to you, be careful about who you share it with and how you share it with them. Think about what would happen if the person you're thinking about sharing your story with got ahold of your diary somehow. How would you feel? Would you be OK with it? Would you be OK if they said you were wrong about a bunch of things? People who are asked to critique stories will go through and look for the parts that they think are wrong and bad, and tell you about them. They might or might not suggest how to fix them. If it's a really sensitive subject that you've been writing about and you would hurt all over if someone told you that any part of it was wrong, then you may not be ready to have the piece reviewed by someone who will criticize. It's OK to not want a critical review of something, especially if you're just starting out and building up the courage to write. Ideally, ask first if the person is OK with you showing them some writing and them giving feedback of the type you want -- and be sure to specify exactly what kind of feedback you want.
Showing it off in public is inviting criticism. If you can't take criticism, don't share it in public. There are many ways to share it that aren't in public, though. You can share it one-on-one with someone; you can share it via e-mail to a person or a group; you can put it up online in a restricted-access area (like a locked, perhaps even filtered post on LJ). If you do share it with someone, let them know what kind of feedback you're looking for, before they start looking it over. It may be that you're looking for someone to go through and point out the strong points without talking about anything you're doing wrong, or aren't succeeding at. It may be that you're looking for someone to point out the plot holes; pointing out errors in your spelling won't do you much good. It may be that you're done with the plotting, and need someone to check for commas out of place and weird formatting. It may be that you need someone to rip the hell out of it and suggest things that could be done better so you can go back through and make serious changes. But know what you need, and let them know what you need, or else you'll probably get them to do it exactly wrong. synecdochic gives good advice on how to get the most out of someone looking the whole story over and making broad suggestions about it.
One of the most stinging quasi-constructive pieces of advice out there is the raw statement "Get a beta." The usual unhappy flailing response is either "I have a beta!" or "I can't find a beta!" Either way, that review means that there are so many technical and structural flaws in the piece that it shouldn't be let out in public on its own. If you can't find someone who is willing to read it and give you constructive feedback, don't worry too hard. Read up on everything you can about the writing process, and see if any of it applies to your own work. Keep writing. If you've reached the limit of what you can do on your own on any one given story, put it down and move on to another. That last story will still be there when you have found someone to look it over, and the things you learn while writing the next piece may be some of the things you were needing to know about the last one.
Spell-check. Always spell-check. There are some things that spell-checking can't fix; there are many things that it can. Grammar-check as well, if the program you're using offers it. http://docs.google.com is a free online document service, and has a built-in spell-checker. LiveJournal has a spell-checker. (It's bad, but it's better than nothing.) Regular use of a spell-checker that flags your misspelled words and makes you correct them will actually improve your spelling, especially if there are a few words that you consistently get wrong. Avoid asking your spell-checker to automatically change your common misspelling to correct it; this will make you more likely to misspell it in contexts where there is no spell-checker.
Be aware of your writing weaknesses. Maybe you have trouble building characters that people believe. Maybe you have trouble with your plot. Maybe you have problems with using the right words all the time. If someone's not willing to give you the full beta experience, but is willing to go through and highlight every time you use a word wrong, take them up on it. Even if they don't actually give you the answer, which word is right, you can look up the wrong word and figure out what it actually means and how it's supposed to be used, and you can probably ask someone else what is the right word that means what you want it to mean that resembles the word that was incorrect.
If you're really serious about wanting to improve your writing on your own, to the point where you'll do anything as long as it's not illegal and won't kill you, look up some of the sites dedicated to bashing the bumbling efforts of new writers, such as deleterius and its ilk. All the quoted bits cited there are probably a Bad Example, but in their deconstruction of the source material, you may find some actual good advice that applies to you. You do need an incredibly thick skin to look at that sort of stuff, though, because a lot of the new-writer/bad-writer bashing can get personal, petty, and mean. You probably don't want to do any interacting with the people there, because you would be a new source of amusement to them and you would probably get flamed up pretty badly. Passively reading the archives won't clue them in to your presence. If you happen to see something you've written there, it really sucks, but getting into a fight with them probably won't solve anything.
If your reviewer suddenly winds up screaming and flailing at you and coming out of nowhere with a very strong and personal reaction that leaves you hurt and spinning, it may not actually be you or your work. You may have just managed to push the hot-button of that particular reviewer, one of the things that is guaranteed to drive them completely insane. Get a second opinion from someone who you don't think has that particular hot button.
Above all, just keep writing. You may only ever wind up writing for your own amusement; you may wind up at the top of the New York Times' Bestseller List; you'll never know unless you keep writing.