I don't know much about Toyotas, but I know how to find cheap parts.
Do not buy cheap Ebay coil-overs to slam your car to the ground. Doing that will most likely make it handle worse. See what other factory parts are an upgrade to your car and look there.
A good place to start may be a Corolla S. If the Corolla S has better a suspension, see if you can find one in a junkyard, then pull the complete suspension (mounts, swaybars, shocks, springs, etc.) from it. If you are lucky, the rims may still be there, too, and they will probably plus size you for $100 or so.
Tires are going to make the most difference. If you are on a limited budget, get the best tires you can, don't worry about R-comps or something like that, yet. Decent all-seasons are fine for daily driving in Florida. If you score a second set of rims like I mentioned above, get sticky tires mounted on those and use them only when you're not on public roads to get as much out of them as you can.
Scour Craigslist and the enthusiast forums. Don't be afraid to drive a bit if there is a good deal a little while away, but combine it with another trip (possibly to a junkyard) if you can. The model specific enthusiast forums will also have a wealth of information about your car. Just lurk, read and learn a bit there to see who seems to know what they are talking about vs. people who just want to make themselves heard. Double check any modification before you do it.
As was said earlier, do the routine maintenance on your car and document it. It isn't sexy and it won't make your car faster, but it will keep it from becoming lawn art and it will help you sell the car if you decide later on that is what you want to do. The best suspension, wheels and tires don't matter for diddly if your engine spins a bearing, right? If you don't know how to change your oil, rotate your tires, do the brakes and a tuneup, beg somebody to teach you those things, because it will save you thousands of dollars over many cars through your life.
Enjoy your car, treat it well, and welcome to the board.