First and foremost, be honest about your gear and experience. If you are in full Karazhan gear, you're ready for some Serpentshrine Cavern and Tempest Keep action. You're not ready for Black Temple and Mount Hyjal, and definitely not ready for the Sunwell Plateau. While you can still submit applications to those guilds, don't expect them to accept you – it's a long shot. In stead, look for well-established guilds that are working on content that your gear is ready for.
Second, face the facts. Some guilds are going to turn you down no matter what gear or experience you have because of your age. That's their prerogative, and don't linger on it. Instead of becoming upset that people are turning you away because you're young, look at it as a challenge to prove that you don't fit the mold of an "emo teenager" or an "uneducated youth."
Third, always spell check. One of the biggest no-nos in any application is spelling and grammar errors. If you are serious about the guild, and serious about raiding, be serious about your application and make sure it's spell checked.
Fourth, don't commit to more than you can do. Some hard core guilds go for four hours a night five or six days a week. Chances are that as a younger person you're not going to be able to do that reliably. While you might be able to sit in front of your computer for hours at a time, if your parents come calling that you need to go take out the trash – you can't tell them no (and get away with it).
Fifth, look for a more casual approach. The hard core guilds are often times going to be unforgiving of mistakes and miscues on your part, and they're not going to want to spend a lot of time training and gearing someone only to have them leave in four months. Instead, look to guilds that are more casual in nature. They will often times provide a better environment for you, including one that you can be in when school resumes in the fall.
All in all, the best piece of advice is to just think and be smart when applying to guilds. Be honest with yourself and them, and give yourself a lot of options. Don't worry if you get rejected – often times the reasons for rejection are completely out of your control. But at the same time take these tips to heart and don't give them a reason to reject your application.
Good luck in your summer raiding!
For those wondering what the picture is, I suggest taking an evening and watching or reading A Clockwork Orange.