Hey,
I've got an old Univega Gran Premio that I ride to work and back everday. My advice to you would be to get that thing tuned up and 'fitted' as much as possible and then ride the heck out of it. By tuned I mean that you need to get the drive train running as smoothly as possible, and keep it clean and properly lubed up. Also, check your brake cables and levers and repair/replace as necessary. If your bike is in tip top shape you'll be much less likely to be unhappy with it as you pass (or are passed by) other cyclists on newer toys. Rest assured it'll be a long time before you are being limited by the bike you ride. Even on climbs, I can hold my own on my Univega, and it's pretty satisfying to out-climb someone on a carbon fiber Dura Ace machine while riding a steel bike with Down Tube shifters. Just because the bike feels like it fits, doesn't mean that it's fitting as well as it can. Find someone in a Local Bike Shop (LBS) to answer some sizing questions and find out what they charge for a basic fitting. Even simple changes like adjusting the seat up/down and forward/backward can make or break the cycling experience for new riders.
The second part of my advice is to ride the heck out of it. You may be told by cyclists to start doing interval training etc. to increase strength, but right now you should stay focused on simply riding, and developing a true love for it. As a 'young cyclist', it's probably more important that you simply put miles on the bike. Ride often in all types of venues: By yourself, with groups, to work/school, with friends etc.
I love riding my Univega, but just like any other bike, if it didn't fit me or run as well as it does, I wouldn't enjoy it. It's a great bike to get into cycling with to find out if it's going to be a long term sport for you. You'll either find that you love your current bike and ride it forever, or decide that you want to be more competitive and upgrade. But don't feel like you need to upgrade anytime soon.
Without knowing what components your bike has, I can't give you more specific advice, so you've gotten my general 'new cyclist' advice. Welcome to cycling and to the forums as well.
-Jeremy