15 year old is getting pretty stale. It's generally a good idea to toss street tires after 6 years.
Depending on how serious you want to get into this, you could either buy the kick-butt awesome tires that everybody raves about, or just buy whatever is cheap and enjoy learning to drive. Or something in-between.
I am running the cheapest tires I could find on my car for both street and autocross. I ~have~ brought race tires to events and swapped tires, but I'm old and rusty and don't care as much any more.