I am currently sat in my office overlooking the Pier Head in Liverpool, the sun is out and I must admit it is a very nice place to be.
Here are my suggestions;
Galleries - Walker Art Gallery, Tate Liverpool (Albert Dock) and The Bluecoat
Museums - International Slavery Museum (Albert Dock), Maritime Museum (Albert Dock), Museum Of Liverpool, The World Museum
You can check out both Cathedrals, both very different but worthwhile.
The Wheel (100 yards from Albert Dock) is worth a go, excellent views
The Mersey Ferry is a bit of an institution, the 60min round trip gives river views and a good deal of maritime history as part of the tour
If football is your thing check out the Anfield stadium tour (you can do Goodison Park but there are less trophies to look at!!)
You can get an open top tour bus in front of the Liver Buildings which will give you a decent idea of what's going on in the city
Of course Matthew St for the Beatles history (and museum at Albert Dock), Check out Blake's at the Hard Days Night Hotel, a great spot for food and drink.
At the Albert Dock you have a variety of very good restaurants, I like the Blue Bar and Grill
Shopping - L1 is ok for a couple hours, plenty of food and drink options
You may decide to venture out to Chester (less than hour on the train)
Crosby Beach to visit the Gormley Statues and watch the ships come up the channel with an ice cream!
Southport is a lovely seaside town (40 mins on the train)
Hope you enjoy, the city really has come on in the last few years,