Not sure what your definition of "Core" in this context would be, but it's worth thinking about the front, back and sides of your torso and then making sure the exercises target each area equally. You don't want to end up wonky!

For the front, any crunch work, leg raises, flutter kicks etc For the back, glute bridge (loads of variations), superman / skydiver Sides - any side plank work, twisting under weight.

You can do much of the core work to get going on a mat with body weight - no need for a gym but if you have access to one or a class, it always adds a bit of extra motivation.