Sounds like your fuel pump is on its way out… The fuel pump is like any other electric motor, it has lots of little windings. One of those winding is bad, and when the pump stops on that winding it can’t start spinning again. The turning it on and off, gets it to kind of jerk past the bad winding to a good winding and the pump starts spinning again. At least that is what it sounds like from your description.
The test would be to put a fuel pressure gauge on the car when cold. See if the fuel pressure comes right up when you turn the key (which it should) or if it only comes up after you turn the key on and off a few times.
After thinking about it, it MAY also be a bad ignition switch, but I a really leaning towards the pump.