Hi, how is everyone at brickboard!

The other day I was driving home from school, and my car stalled and broke down with smoke coming from under the hood. As it turns out my car ran out of radiator fluid (either through a leek or otherwise) and part of the radiator literally melted; as a result the engine overheated. I limped the car home by stopping every few miles and filling the radiator with more water (that would rapidly leak out of the hole).
I replaced the radiator as well as the fluid.
Now here is the problem. The car has a very difficult time starting, (with muffler sounds and a lot of shaking, as if its not getting gas,) but if you pump the gas pedal, it will start up after a few seconds. Once the car is running for a minute or so there is no problem at all, the car drives fine, and I have even taken it for long drives without issue. But if you turn the car off for more than a minute, there is the same problem trying to get it started again.

I am puzzled.
Any help would we greatly appreciated :)