I have a fule problem

Its an automatic with a 2 litre and 117000 miles. I don't know the work history because we just bought the car. The problem is with a cold engine, If it was recently turned off, just for a few minutes, it will start up OK. My wife drives the car, and says the trick is to turn the key on 3 or 4 times before running the starter and it will start.

I replaced the fuel pump and filter and I still have the same problem. The car won't start until the 4th try. Sometimes, after cold starting, the engine immediately revs up for a second, then chugs a little for a second, before it idles normally.

Sounds like a leak. Fuel pump?--> Maybe. Injectors?--> Maybe. If it has a pressure regulators not a returnless system. Pressure regulator ---> maybe. Very interesting. Any pressure reading n then PROCESS OF ELIMINATIONS by crimping 1 section at a time to see if pressure hold up to validate the leak yet. Huh?