The best place to start is the Old Market Square as everything leads off of there, I love the big water feature in the square. Parking is a bit of a pain but there is the trams, buses and trains to get into the centre if you don't mind taking public transport.
If you come across The Robin Hood Legacy, just don't go near it. My husband and I tried it and wasn't impressed in any way.
To be completely honest, Nottingham Castle is more like a huge manor house with short castle walls around it, it's not like the films but still great to walk around to see the exhibits on the local history. If you walk around the outside of the gates and castle walls, you can see the entracnes to the tunnels that lead up and into the castle. Robin Hood would of used some of them when sneaking into the castle.
If you are looking for a pub with a lot of character, check our the Ye Old Trip To Jerusalem, part of it has actually been cut into the rock under the castle. The front looks like a normal building until you go inside, it’s like sitting in a cave with a pint haha. It also boasts as being the oldest inn in England as it was built in 1189.
There are a lot of little streets with shops if you want to go on a shopping spree or check out the beautiful architecture of the old buildings. Some of them are so old and the detail that went into those buildings is nothing short of incredible.
Red's True Barbecue is another great one, the staff are great, the drinks are super yummy, food is delicious and the decor is out of this world.
If you just want to sit down with a hot drink and chill out, Find St Peter's Church as they have a lovely cafe not far from the Market Square.
The National Justice Museum is a brilliant way to spend a few hours, they have great exhibits and can take you to explore the City Of Caves.