Starting problems HELP
 

I recently bought a 1987 supra turbo targa with 58,000 orginal miles for $4200 totally rust free and babied, including a new head gasket which I am glad to see with all the head gasket problems with these cars. As far as I am concerned I stole it. I had to drive to Oklahoma from Chicago to pick it up but it was so worth it. The car is totally original and seems fast, but then again this is my first turbo car. My problem is that the car doesn't want to start especially after sitting for about 15-30 min. after being driven. It cranks over fine but just doesnt want to pop off. If I keep the key on start it will eventually start without any gas but not without turning over for at least a good 3-4 seconds and I am affraid if I keep this up I will eventually go through a starter motor. If I crank it over for 1-2 seconds, stop, then start turning it over again it starts better but then idles funny like it is flooded. You know hit the gas and it almost stalls, but if I kinda of feather the throttle I can get through it. I have heard Throttle PS, cold start injector, or fuel regulator. Does anyone know how I can narrow this down. Please help!!!