I would keep it on the counter. Every day, mix together equal amounts BY WEIGHT of starter, flour,
and water. Discard the rest of the starter or use it. (It's wasteful, I know) It should get bubblier each day and eventually it should double in size each time you feed it. Once it is that active, you can refrigerate it and then go through the feeding once a week.
If your starter isn't quite active enough you can still use it for flavor, adding a pinch of yeast for insurance. That's what a lot of commercial bakeries do.
I find with sourdough the bread takes longer to rise no matter what. Forget the times the recipe says and judge by volume instead. It's worth the wait.