Nottingham is a really vibrant place to live. Big city, but also very close to lots of countryside for nice walks. Only 1hr 40 mins fast train straight into St Pancras so great for getting down to London quickly. It has an amazing tram network (Which goes straight through the railway station) and the buses are great too.
Neighbourhoods closer to the City that are considered nice neighbourhoods are Mapperley Park and The Park. You can walk into town from both. MP is next to Sherwood which as previous posters have said has a really arty vibe, with lots of independent shops and cafes/restaurants. Local primary schools have improved dramatically over the last decade and are now all good or outstanding. Mapperley itself is nice too, great schools again - good or outstanding on the whole, nice high street. Slightly further out. Woodthrope nice too.
Beeston is on the tram link and has a lot of new development happening over there. A little further out past the University of Nottingham, but doesn’t take long to get into town as it’s on the tram.
South of the river you have West Bridgford, which is very desirable. Very yummy mummy-ish and can be a bit footballer wives ish sometimes. Some really lovely houses, expensive, but it’s very busy. Lots of traffic around Central Avenue (where the shops/restaurants are). As previous posters have said, the outer bits Lady Bay, Compton Acres, Wilford Village are a bit cheaper but still nice.
Having lived in London for 15 years, I really like living here. Everyone is very friendly and down to earth. Much calmer way of life and really appreciate the facilities of a big city and easy access to countryside.
One slightly confusing thing is Nottingham City vs Nottinghamshire County Council, you could end up living in either, but it’s basically one big conurbation until you get to the outskirts and it changes to countryside and villages, and then it’s NCC. School districts could be in either City Council or County Council.