I can't comment on the uni as I'm still in year 13 but I live near and go to school in Bath so I know about the city pretty well.
There aren't many clubs [if any..] in town but there are a lot of pubs which are good for going out - or so I find with my group of friends anyway. There is then the issue of them having the odd group of sixth formers sat around but what can you do eh? I've found they do get pretty busy as well and you could struggle to sit down at certain times especially when they shut the outside parts [Pig and Fiddle is one of the worst for this].
There are affordable restaurents like Pizza Express and Wagamamas around, a cinema, a Wetherspoons and a few more things. You do have to make your own entertainment a lot but Bath is a really nice place to be if you're willing to put the effort in. There is no getting around the fact that it is very much an upper-middle-class city but I don't notice it too much now I'm used to it [although I was in Waterstones today and the people behind me were arranging a polo match...] so I guess you get used to it. Bath is a 'going for coffee' city with 3 Neros, 2 Starbucks, Costas and God knows what else in the town centre.
The uni is known for its sports and there are quite a few people from my sixth form who go up for training and whatnot who say it's all very active with students so no worries on that front!
Uni students are frowned upon by a lot of the 'typical Bath' people who think they're messing the city up and want them gone and as a result you don't see too much student-y stuff around town.
Hope this helps a little!