In reply to aln:
The city walls are a great way to see the historic city.
Museum gardens and the abbey ruin are nice. Maltings and Minster in both interesting local pubs.
Railway museum is cool if you like machines, currently have to book but free.
Greys court behind the minster is a lovely building, nice quiet place for a drink or cake. The treasurer's house garden nearby is nice.
The House of trembling madness on Stonegate is a tiny bar in a medeval attic, more interesting than its mansion based namesake.
The Blue Bell is intteresting, tiny, old and charming.
Fossgate and Bishopthorpe rd are both busy little streets with lots of independant shops, bars etc.
Exploring the little lanes (Snickleways) and backstreets in town rewards with interesting finds. Barley hall being worth seeking out. Loads of independant cafes, eateries and bars tucked away off the beaten track, finding one you like by chance is half the fun.
Marks and Spencer cafe, top floor has a great city view, worth a look especially if it's raining.
Yorkshire air museum at Elvington is cracking but it's a few miles out, there's probably a bus but it'd eat half a day, only worth it on a short trip if you're really into aircraft.
Station tap in the station is a nice bar.
Brew York, brewery and tap room for modern beer.
Holgate windmill is interesting, oddand scenic if it's open. Half hour walk out of town (less from railway museum) but past good pubs (Volunteer arms, a cosy local and the Fox, big garden. Both do good beer).
Middle feast for a kebab to enjoy sober or the food market off the Shambles for other street food.
Margret Clitherow's shrine on the Shambles is an oasis of calm amid tourist chaos if it's open pop your head in. Grim backstory.
Clifford's tower is closed for work.
Various cinemas and theatres in town, not sure what's on but if that's your thing you're well set and noneed to go to the ringroad multiplexes.
The uni is a mad mix of brutalism, ducks and medeval if odd architecturr is your thing. It's about 45min walk out of town or there are busses.
Free walking tours from the art gallery are supposed to be very good. Paid ghost walks from everywhere, look like fun but never done one.
The little museum under tge roman bath pub is quirky and would make a memory.
Minster is spectacular but pricey, the basement and roof are worth tge extra imo, particularly the basement treasury, roman stuff and mad Arup 60's undrrpinning if you like engineering.
No idea what the live music scene is like mid/post covid.
If you have kids Rowantree's park is nice and close to town, access via riverside walk might be disrupted by works.
Enjoy, jk