I am very familiar with B'ham, having lived there as a child and I am a regular visitor to see friends and for work ... B'ham is a great place, but gets a very bad press in the UK. Admittedly, it's not intially the most attractive of UK cities (especially when viewed coming in by train from London), but the centre does have some great Victorian architecture, exciting new architecture and attractive suburbs. It has a good range of cultural and shopping faciliites and a lively student scene, with two big universities. It's also a diverse city, with a large non-white population. The local economy is good too. Outside B'ham are some great towns and villages, set in beautiful countryside. So, you have made a good choice!!!
As for places to live, there are lots of new apartmets in the city centre or try the more established south west suburbs of Edgbaston, Bourneville, Kings Norton, Kings Heath and Selly Oak (student area). Less expensive areas are to the north of the city centre. To get an idea of prices, maybe look at the local newspapers - the Birmigham Post and the Birmingham Evening Mail.
Good luck!