Where to start troubleshooting this problem

Car will not start when you run it for a short time turn it off for more than a few minutes and try to restart. Also will not start if you try to crank it over and don't and then try again. It will not even try to crank the engine. On long drives there is never a problem. This happens only on short trips and if you don't get the engine started the first time you try to turn the engine over. The next day you go out and it starts.

So battery is new. When I turn the key on in ignition all normal lights come on and aren't dim. Solenoid? Starter? How to test, voltmeter? Any other ideas of what it could be. I hate to replace these without knowing for sure they are the problem. I have already taken it to a repair shop last time this happened. Unfortunately by the time they are able to get around to looking at it, it is running again. So then it makes me wonder is something shorting out? They say that they can't test while it isn't happening, and I appreciate that they didn't just replace at random and charge me. It is just so frustrating. Last week it left us stranded 45 minutes from home in the middle of the night, all 5 kids in the car.