I've just found the "illuminate" spell in the playground and I don't know how to equip it. There isn't an equip button or anything so how do I use this?

It is a bit wonky, but since there's no equip button that appears, you have to click the desired ability button (should highlight blue) and then click it again, while it's still blue. That should toggle its equip/active status.

Illuminate's brightness also scales with your magic attribute, so it may not be that noticeable at first.