If you don't have much track experience - say, fewer than a dozen track events - then I recommend using whatever tires you are using on the street - whatever size and type of tire. You won't be using anywhere near your car's capabilities, so there's no need to worry about it.
Once you start building up a lot of experience, then you'll probably want to consider making the jump to R compound track tires, and get a separate set of wheels to use. At that point, you'll want to consider what type and size of tires you want before you buy the wheels. Track tire availability varies depending on the tire size (and, of course, wheel size).
I'm using R compound track tires (now Toyo RA-1, formerly Yokohama A032R) on the stock 15"/16" wheels in the stock '91-93 sizes. They give great performance, much better response than street tires for track use. They also fit inside the car, so I can drive to the track on my street tires and swap at the track. And you can get used sets of the wheels for almost nothing ($100-200 per set). However, those wheels cannot fit on the '97-05 NSX or earlier NSXs with big brake kits.
With both brands of R compound track tires, heat cycled but not shaved, I'm typically finding that they last 1300-1500 track miles plus another 400 street miles (between the track and the hotel). Unlike with street tires on the NSX, the fronts do NOT last 2-3 times as long as the rears.