I second the recommendations for Galleries of Justice, Broadway Cinema and Nottingham Contemporary. There is a private members-only library in the city centre called Bromley House, which is a hidden gem, and they do guided tours and occasional open days, so I’d contact them in advance if you’d like to look around.
For lunch in the city centre, I recommend Baresca (tapas) which is near Bridlesmith Gate (I’ve seen someone else recommend this for shopping - it used to be full of independent shops but is now mostly chains, although I recommend The Tokenhouse for stationery, cards and gifts). In the Hockley area, nearer Broadway Cinema, I’d recommend Sexy Mama Love Spaghetti (Italian) or Edin’s Deli.
For a drink in a more unusual building in the city, try Pitcher & Piano (near Galleries of Justice) which is in a sensitively-converted church, or Malt Cross (near Bromley Library) which is in an old music hall.
If you fancy getting out of the city centre, I’d recommend Ruddington which is a village about 20 mins South of the city (regular buses from the centre). There is a large country park which was originally a munitions depot in WW2 and has been developed with some great walking trails and a lake. In the village, there are some lovely coffee shops/pubs for lunch (try The Old Bakehouse or Jasper’s cafe, or the White Horse pub). Then I’d spend an hour or two in the Framework Knitters Museum (another hidden gem, and slightly quirky but if you like crafts think it could interest you). Followed by a ramble around the independent shops in the village (including a wool shop, keeping on the knitting theme!). If your family wanted to join you there for dinner, the Ruddington Arms regularly wins awards for best gastropub in Notts.
Enjoy your break!