The London Underground system is really easy to understand. (Even American tourists can work it out, so it must be simple!). You can buy a Travelcard, which gives you a day's unlimited travel within central London (by both bus and underground) for �6.60. If you don't need to travel before 0930, you can buy the off-peak version for �5.10.
The nearest underground station to the Palladium is Oxford Circus, which is on 3 different Tube lines (Central, Bakerloo and Victoria), so you can get there easily from many parts of London without even changing trains.
There are loads of cheap, but good, hotels in London. For example, the Umi (in Bayswater) charges between �35 and �47pppn for room only. It's only 5 minutes walk from Bayswater and Queensway underground stations. (Queensway is on the Central line, for direct access to Oxford Circus).
The Cecil Court Hotel (close to Lancaster Gate tube station, on the Central LIne) at �25pppn is also worth considering You could also take a look at the Marble Arch Inn (�32.50 to �35pppn) which is close to Marble Arch tube station on the Central Line (although you might like to walk to Oxford Circus, straight along Oxford Street).
All of those hotels are within 5 to 10 minutes journey time, by tube, to Oxford Circus and the theatre is just around the corner.
You can find those budget hotels, and many more, at
http://www.hostelworld.com
(Use the 'Additional search options' to select 'Budget Hotels' and 'Guesthouses').
Chris