There are several small outfits that sell used but still very serviceable full on slicks or R rated compound tires. SOme are advertising in Grassroots Motorsports magazine. You could probably Google used race tires. Don't go any wider than stock.
I would use 16" wheels. 15 would be good for autocross, but you need more tire for racetrack use. There are lots of 16" tires out there.
Wheels need to be true and straight. A damaged wheel, even slightly damaged or warped, can cause lots of problems at high speeds. Get a good set from Tire Rack, and buy used R compound tires. I think Tire Rack has a good tutorial on tire nomenclature, and that will help you immensely.
Proper inflation is also critical. Start a little low on pressure, as the air in the tires expands as it heats up. Get a good tire pressure guage and measure pressure and adjust after every run. Tire temps are important, too.