Being on a big bird is just awesome! To make it more enjoyable, reserve a seat next to a window, and away from the wing..this way you get to see what's it like to see the ground from above. I love flying.
So, you want to go on a trip, decide where you want to go, from what airport to what airport you will be traveling. Know that there are rules and regulations for you to follow and will make your life easier.
For starters like you, take a local trip. Say, if you live in Los Angeles you can fly to San Francisco on Southwest Airlines, Alaska Airlines, American Airlines, Continental Airlines, and other domestic flights. The flight is 1:00 hr, so you'd be there in no time. And I must say, San Francisco is a place that is perfect for traveling.
Take only what you need, for the lenght of time you will need it for. Have your legal documents, such as a driver's liscence, student ID, credit card, anything identifying you. Be sure to book your flights 14 days in advance or more to get a reasonable priced flight.
If you are not sure which flight is best for your destination, I highly suggest you go to to Orbitz.com and put in your airport of origin and your destination, and it will display the airlines and prices that fly to where you want to go. They're very good in prices.
Normally I call the airline directly, because I know where I'm going, but sometimes I use Orbitz.com. Once you do that, I suggest you get into their Frequent Flyer Program, most airlines have it, and you will earn points for every flight you make. It's free, and is not a credit card. After certain amount of flights you take, you earn free hotel stays, free car rentals, or free flights.
I hope you enjoy your first trip on an airplane. Cheers!