It's been (yikes!) twelve years since I was there, but it was most interesting! The most obvious suggestion is the "Magical Mystery Beatles' tour." I know it sounds cheesy, but it really was interesting to see many of the places, such as "Strawberry Fields" that inspired the famous songs. Also, by the water (please forgive me; I'm a bit tired and can't even remember the silly name...), there were some very cool shops and at least one museum. (I want to say "wharf" but I don't recall exactly..)

Unless things have drastically changed, I think you should be aware that there are some dangerous areas in Liverpool. When I was on the Beatles' tour and we were seeing where Ringo Starr grew up, a group of ~8 young boys were making very obscene sexual gestures when they saw women inside the van. The driver told us to keep the doors locked and to roll up the windows. So, I'd make sure that anywhere you travel by yourself, you find out from others that it's relatively safe.

Oh, and although it may not be as noticeable to you as an Irish person, but the Liverpoolian accents were t*h*i*c*k!!