I agree with sparrow...and fully disagree that stability tools such as balls, BOSUs, Sitfits are "overrated at best" or a "complete waste". You can certainly progress to using these toys, and they are very challenging and provide great results if used properly, of course. I do agree however, that a variety of exercises can be used to work the core that do not necessarily involve using the aforementioned tools. A very simple way to work your core without using any of the said tools is by challenging your proprioception. Perform exercises with your eyes closed, one eye closed, or looking up at the ceiling instead of at yourself in the mirror. This takes away your peripheral vision and eliminates your ability to see what your arms/legs are doing.
As mentioned by trainerty, anytime you are without support, you will be using your core. Try to do exercises standing up instead of sitting down. Instead of doing bent-over rows with one knee and one hand on a bench, simply do them bent-over, on two legs, and eventually on one. See how it goes?
If I have to say anything it’s a) use your imagination, b) look around…if you are lucky enough to be surrounded with some experienced and competent trainers, you will learn a lot from seeing what they do in the gym and asking them questions, and c) play safe!
**Proprioception: the ability to sense the position and location and orientation and movement of the body and its parts