The most important things to test for are ammonia, nitrite and (to a lesser extent) nitrate.
If you are cycling, whether with or without fish, test once a day for ammonia and nitrite. Same applies when you've just added new fish. Also if your fish seem ill, tesing for ammonia and nitrite can help identify the cause.
The rest of the time, I just check nitrates once a week because high nitrate can indicate that you're not maintaining your tank regularly.
There's a whole lot of other things you can test for aswell but I don't. I think if you've just set up a brand new tank and are choosing which fish to put in it's a good idea to test both pH and hardness to see what fish your water is suitable for.