I only spent one day in Manchester because there isn´t that much to see. I had dinner in Chinatown (which is btw the third largest one in Europe) in a Thai restaurant, which I can totally recommend. I have to admit that it may look a bit decayed from the outside but once you´ve entered you feel like you´ve just travelled 10.000 km around the word and the food is delicious.
I would say John Rylands Library is a must-see. It looks like a church from the outside and inside. When you are in the large reading room you feel like you´re in Hogwarts - at least I did - surrounded by thousands of old books that you unfortunately can´t touch...
For the Beatles fans The Beatles Story is a place to go, where you can see everything from the beginning until the end - if you don´t already know. Also Mathew Street is all about the Beatles with stores and bars. Penny Lane and Abbey Road however, were quite a disappointment.
Keep traveling xx