Liverpool city centre isn't big, so you can go anywhere, but Lime St isn't the most salubrious area!
Lots of buses and trains if you want to go a bit out of town, but most of the main sites all walkable in city centre.
If you like culture, there are 2 cathedrals, Walker Art Gallery. Liverpool Museum, walk around the Georgian Quarter.
If you like the Beatles , there isca museum plus bus trips out to Penny Lane and Paul's house (National Trust).
No end of bars and clubs - try around Bold Street and Concert Square.
Royal Albert Dock and Pier Head - more museums, river, restaurants.
Get the bus, train or a taxi out to Crosby to see the Anthony Gormley statues.
Theatre - Empire, Playhouse, Everyman, Shakespeare North in Prescot, Epstein.
if you're a shopper, Liverpool One - large shopping complex.
Just Google any local tourist website! There's enough to do to fill a month, so do plan a bit in advance.