I reckon you’ll be bored by the end of day three, but here goes!
Galleries of Justice, you can get it with a joint ticket to the caves which were a bit naff last time I went, I think it depends on the quality of your guide.
Apparently the ghost/cave tours where they take you to some of the ‘private’ caves under pubs and whatnot are better.
You could go up to Sherwood forest and check out the oak.
Nottingham Castle is, frankly, not worth the admission unless something else is on in the grounds. Spoiler: There is no Castle, just an outline of bricks where it used to be (the Cromwellians razed it, as it was where Charles I raised his standard). The museum in the Castle is pretty bad too, there’s a reasonably interesting history to Nottingham (see above, and what with the lace trade/industrialisation/decline) but the museum feels like it’s filled with random cast offs from the V&A, it’s really lacking a thread to the exhibition.
Eating, off the top of my head.
The Larder on Goosegate is good for British food done really well.
Iberico is good for tapas.
Shanghai Shanghai is ok for noodles/chinese
Miss Korea is tasty for Korean food, so I’m told.