Like snaefell052 I'd also recommend the Museum of Liverpool as there are several Beatles artifacts in there including the stage from St Peter's church hall:
Plus, The Beatles show short film: liverpoolmuseums.org.uk/mol/visit/galleries/…
You need to walk past the Museum of Liverpool to get from the Beatles Story site at Albert Dock to the second Beatles Story site at Pier Head where there is also a special exhibition: http://www.elvisandus.com/
If you want to go to a pub where the Beatles used to hang out then go to the White Star at the bottom of Mathew Street (and around to the right).
In the lounge room at the back of the pub is where the Beatles used to have a pint with Allan Williams, and there are several interesting Beatles artifacts framed on the wall - not forgetting the brass plaques of "John", "Paul", "George" and "Ringo" on the bench. That's where they sat of course!
If you want to meet Allan Williams then head back up to the Grapes on Mathew Street. If you see an elderly, but very talkative, gentleman with crazy white hair sat at the bar then that will be Allan Williams (shhh, don't tell everyone!). But if you get talking to him about the Beatles and want to hear his stories then expect to buy him a glass of red wine. (He's been perfecting that little trick for years and I've seen him play it in the Jacaranda before it sadly closed - but it's worth it !!)
In the Grapes you can also have your photo taken next to the plaque "This photo was taken here". You need to work that one out yourself though ! ;-)
If you have any spare time (you have Sunday morning free don't you?) then I would highly recommend going to the Walker Art Gallery (next to St Georges Hall and Lime Street station) to see the Rolf Harris "Can you tell what it is yet?" art exhibition:
The Walker is open at 10am and it's free entry.
And as purdydog mentioned above, go to the Hard Days Night Hotel at the top of Mathew Street and have a Beatles themed cocktail at Bar Four www.harddaysnighthotel.com/bar-four.html