street tires on the more aggressive side are fine. e.g Metzeller M3, Pirelli diablo rosso corsas, dunlop Q2's. I'm not a michelin or bridgstone fan, but they have options as well. Personally I really like the Q2's for entry level track performance. Q2's are better if its hot, DRC's a little better if its cold IMO, but in terms of value, I'd go with the Q2's hands down, they are a good margin less than the DRC's. You can ride them for quite a while, until your mid-level red group, then you'll need something a bit more. You don't need warmers on street tires, but to prevent cold tearing, cold tire crashes, and not wasting 2 laps every session warming them up, they are a good idea. Most don't start out with them, usually something people buy down the road.
I don't know anyone who rides shinko's at the track. They are cheap because they are older technology, significant advancements have been made since. Some of the drag guys use them, but I would not recommend that route for your purpose.
