Why are we having random starting problems with the ignition

What is your question?

Recently we have been trying to diagnose why it sometimes starts right up but other times will get not even a click when turning the ignition. We had AAA come out and they said it was a bad battery and they replaced it with a new one. Thinking we had the problem solved, it did it again last night but after several attempts at turning the ignition, it finally started and I tried again this morning and it started right up. I am just trying to narrow down the problem so, we are not spending all kinds of money on various parts or at the mechanics. I am thinking maybe it's the ignition switch or the starter solenoid.,,,,not sure Thanks in advance.