What's it take to "Track" a car?

All depends on what type of track racing you are doing, but for your typical HPDE day, just the normal maintenance items really need to be done. I'd cut your teeth on the street tires you have first, then if you like it, look at upgrading. You don't have to go crazy either, we run Falken Azenis on our lemons car and they work just fine. Not the fastest tire, but they are a good value IMO. Make sure you have:

1. Fresh brake fluid (I like to run the synthetic DOT3 Valvoline, works just fine for and is cost effective to change after every race).
2. Good brakes. I would change over to a more track oriented pad just for safety's sake. I had good luck with the EBC yellow stuff pads on my AX Cobra car back in the day.
3. Good helmet (I run a pyrotect helmet - SN2015)
4. Decent pair of shoes (driving shoes are the best, especially once you try to heal-toe and rev-match) but any sneaker will do.
5. Decent pair of gloves. I like driving gloves since they help me grip the wheel better, not necessary, just nice.
6. Whatever safety items the track/organization will require you too have (cotton undershirt, ect ect)
7. Make sure all your suspension items are in good working order. You don't need anything fancy for your first couple times out on the track, just make sure nothing is loose.
8. Small fire extinguisher in the car (5lb)
9. Make sure your battery and any other items in the car are tied down securely. Remove the floor mats before going out.
10. Fresh fluids (Engine oil, rear, coolant). I usually just run straight water on the track with some water wetter for corrosion prevention purposes.
11. Some way of holding yourself in the seat. If you have side bolsters, great, but you will still move around. A harness is best, but the old twist the stock seatbelt to hold you in the seat works well.

Best advice I can give you, have an instructor go out with you to give you pointers on your first event. They can teach you braking zones, lines through the corners, surface changes, ect that would take you many track events to identify. You're not going to be the fastest the first time you go out. Start off slow and learn the car, put it into corners hard (when you know you have an out) and judge how the car reacts. If you have the opportunity, go do an autocross event first. That will really teach you the limits of your car at a much slower speed. Take your time and build up your confidence and skill.