I'm a stay-at-home mom; what are some things I can do to engage and further my intellect?

Just revive something that you were keen on doing but didn't have the time to. Like I started with blogging and freelance writing. It honed my writing skills. There are so many things that can be done. You can take up learning a new language or instrument, singing, writing. If you are inclined towards art and craft, then take up embroidery, stitching, painting..