Check the coolant temp sensor. It sends temp info to the computer that controls the amount of fuel the engine gets as it is warming up. Had mine fail, and the computer thought the engine was always hot, and did not enrich the fuel for the cold start up. Simple, cheap and easy to replace.
For simple diagnostics, if the car runs better and better as you manage to get the engine to warm up, this becomes a high suspect. Be careful not to replace the sensor for the temp gauge, they are different.
My symptoms were, almost would not start when cold outside. Would sort of catch every now and then. Once caught, just barley sputters, and required much open throttle to keep it running. Kept on stalling. Persisted with starting and as engine slowly got warmer, the engine slowly stared to run better. Eventually ran perfectly. Once cooled off, back to square one. Replaced coolant temp sensor - bingo, all better.
Mike