Unless you're afraid of putting some reasonable wear on the expensive tires on the car, you shouldn't worry about that yet. Odds are that size wouldn't be any faster that what you have now, unless you paid enough to get sticky tires.
A better investment would be seat time. Get the schedules for the year of the 3 or 4 closest other clubs, put it all on a calendar, and see which ones you might be able to make. If you must buy tires, get something affordable in a stock size and try to destroy them with as many events as possible. You'll probably run a year or two before tires are relevant.
An experienced driver will gain a second or so from sticky tires. A good launch is worth a few tenths of a second. Experience is worth several whole seconds. A rookie can learn bad habits with great tires, while things you learn on bad tires will still apply later to sticky tires.
05 ZX4 ST - parts donor. Daily driver until a rock found the oil filter and locked the motor.
08 S Coupe - autocross toy. Less weight and more tire.
06 Ranger - lowered commuter and parts hauler.