Car will not start

I have this car for three days now and it ran awesome for the first, but now it will not start in the morning until I wait for four to five minutes.

When I plug the key, everything lights up but no "check engine" light. The fuel light comes up and disappears after some seconds.

I try to crank with the key and it tries to start but nothing happens. I remove the key and try again with the same result.

I wait for other two minutes then I turn the key on and the "check engine" comes (blinks sometimes for a second or two) I crank but it does not start. I remove the key off and wait some second, then tries to start it and it starts then shuts off immediately. I wait more and try to start then it starts with some hesitation. I give it some power to keep it running and warm up.

Then the check engine light will be on all the time, until (i do not know if it is accurate) it gets the proper heat and the fan starts.

While I am driving it sometimes turns off and the check engine comes on. I put it on (P) and starts it again, and the check engine light still on.

Any advice?