I'll take a whack at it and say there's moisture in the lines. I'm purely
guessing here, but I think that water separates from gas when the car is
sitting, and when things get moving the problems start. My old base model
Daytona used to do that until I put some water remover in the gas tank. The
problem stopped after that. Of course, the car had many miles at that point
and snake oil didn't look so bad. I'd think hard about adding anything to
the oil or gas (or coolant) before doing it.

Second gas would be shite gas. It happens.

JW
2003 Neon SXT Auto
2004 Neon SXT Stick

Since you didn't mention an engine lite being on What you may have is the
fuel draining back to the tank, You have 3 seconds to start the engine until
the shut down relay shuts the fuel and spark off until you turn off the key
and retry it. If there is no fuel in the lines at the time you try to start
it, it wont start until you do it a few times. Now if it starts up good cold
that wouldn't be your problem. It could be an out of calibration coolant
temp sensor.
My opinion only

Glenn Beasley
Chrysler Tech