I'm guessing, Molly, that you're considering possible university choices here. Just about everywhere on your list has a fairly large student population, with the life-style that goes with it.
Brighton is lively, vibrant and packed with young people. As well as its two universities (Brighton & Sussex) there are loads of schools teaching English to foreign students, so the city exudes a sense of youthful fun. It's got plenty of good shopping (especially around the North Laine area, where you'll find lots of quirky independent traders) plus loads of pubs, restaurants and nightlife.
I've attended a conference at Exeter University. Even though the campus seems to have far more hills going up than down it's a friendly place, with a city centre which has a 'cosy' feel to it (rather than Brighton's 'brash' aura). It's much quieter there than it is in Brighton, with far fewer off-campus attractions for students.
I don't know the other cities particularly well but I'm aware that Southampton is very much a 'student city' (even though the city centre itself is rather dull).
Cardiff must rank second only to Brussels as the most boring capital city in Europe.
Bangor is dominated by the university, with around 10,000 students and only about 14,000 other residents. Consequently, Bangor doesn't have a great deal to offer outside of university activities.
I don't know enough about Plymouth or Portsmouth to comment on them.
Chris