The Castle is interesting - it's not really like a castle, more like a big old house, but it's on a huge rock with good views - there is also a great pub just below the castle - also, go visit the Lacemarket - a network of old streets with little shops and cafes - all in the city centre - there is a good tram network too to get you from the train station - Nottingham is a good place at night too - I've found it to be pretty relaxed and friendly - try the pubs along the canal. Have fun!
PS - don't go to the Robin Hood museum - it is NAFF!