The things that get used most often in our house, by me are:

-flashlight
-screwdrivers, with various heads
-wrench and vice grips
-hammer
-various nails and screws
-duct tape and electrical tape

It's not much but I can open stuff to change batteries, fix stuff by banging or screwing it back together, and hang stuff.

My husnad is much more handy than I am, so he's got a huge array of tools nad gadgets. It all depends on your skill and interest level. I'd rather call a specialist like an electrician or plumber when things go really wrong.