Depending on the organization, they will only require 2 piece textile or leathers that zip together, (no mesh). I've seen riders on the track wearing 1 piece Aerostitch suits. It depends on the school, so call around and see what they say. As long as you have a full face helmet, and full coverage boots and gloves, you should be fine. if you have any questions of course, I hope you'll let me know.
I Don't Have a Track Bike
I didn't either. And I still don't. I took my street bike to the track. Back in 2011 it was my SV650s. Last year, it was my beloved Street Triple. Although the Suzuki was a better bike on the track, due to the aggressive riding position. But I've seen all different types of motorcycles on the track, and ask anyone who's been to one they'll probably tell you that they've seen more than your typical "track bike".
If you don't know how to ride within your limits, then yes, you'll probably destroy your bike in the process. Most people riding in their very first track day naturally go a lot slower and ride a bit more carefully because it's intimidating and a completely unfamiliar place. It'll take you awhile to get used to the feel / idea of being on the track too. If anything, it'll probably humble you quite a bit and really show you what you don't know how to do.
But some schools also offer rentals like CLASS, which offers Honda CBR 300 and 500s. And Yamaha Champions offers Yamahas. So you don't always have to ride your own bike. Many track day organizations offer bike rentals, you just have to ask.
Something else to consider are track days that are specifically focused on street riding like Z2's Road Rider 2.0 Course. The curriculum is totally different from a traditional track day too. There are lots of other courses like this offered from other organizations as well, like Yamaha Champions "ChampStreet" course.