First of all, the Waterfront itself around the Albert Dock is stunning, and a World Heritage Site, so take a look at the fantastic buildings around there. On the Albert Dock itself, you've got the Beatles Story museum, which takes you through the fab four's career and is good if you're a fan. You can also jump on the bus for the Beatles tour, which takes you around the places the Beatles met and grew up. Other museums include the Maritime museum which looks at the history of the docks, and the museum of Liverpool life which looks at the history of the city and community. Also from the docks, you can take the 'Ferry across the Mersey' for a boat tour of the area.
In the city centre, there's lots of great bars and restaurants. Recommended is the St Petersburg Russian Restaurant on Duke Street, or if you're on a budget, Caesar's Palace on Renshaw Street sells good burgers and steaks! Also in the city, have a walk around the Cavern Quarter, and take a look in the Cavern Club, which isn't the original but is a close replica! Lots of good shopping too, particularly the new Met Quarter shopping centre. Also, on Stanhope Street is Cains Brewery which do a tour which includes a free pint of ale!
If you like Art there are several galleries such as the Tate on the Albert dock, the Walker Gallery and the FACT centre which also shows movies.
Outside of the city, its worth going to Anfield, the home of Liverpool football club, where you can go on a tour and look around the museum and trophy rooms.
Plus there are lots of great parks around the suburbs, such as Calderstones Park and Croxteth Country Park if you want to get close to nature.
A little further out there are beaches at Crosby and Southport, Crosby beach having a semi-permanent art installation of lots of sculpted men along the beach which are very cool and interesting. At Southport you've also got golf courses, an amusement park called Pleasure Land, plus a model railway and other seaside attractions. At Formby there is also a nature reserve ad red squirrel sanctuary.
Also further out is Knowsley Safari Park if you want to see monkeys climbing on your car!
Getting around is fairly easy, with good bus and train links around Merseyside, plus the people are very friendly so just ask if you get lost. Have a great time!