The City of Liverpool offers many tours for Beatles fans, but after reading many brochures and asking for opinions at the hostel, we decided to do this tour. It departs from the port, first stop Penny Lane, a famous area due to the song of the same name. Stops are made at the homes of John and Paul, at Strawberry Fields and other points of interest. The houses are now occupied, so you can't go inside. The tour ends at The Cavern, a pub where the best rock bands of the time played, and where The Beatles played, of course.
It is a bar that is very close to Royal Mile that has live music every day and night. Every Monday there is an "open mic" night where a group performs more or less every hour. In theory they are amateurs but...I have not heard about amateurs being that good in my life. 100% recommended. On the web you can see the schedule and groups :)