Daniel
I could drone on for hours, but I won't. First and foremost, if you take the role on, you will need to change the signatories on any accounts the band has. Obviously you will need to be able to sign cheques, and if they need counter-signing, you will need to get other people added to the account as well. This process can be a bit of a ball-acher, but don't let it get you down.
I spent a year as treasurer of a band in York before I left for pastures new, and as long as record everything as it happens (eg dep players, new music), you will be OK. Shadowing the current treasurer is a good idea as you'll get a chance to see the system in practise.
The best piece of advice I can give is, don't be afraid to ask questions, either of the current treasurer, or other people that might be able to advise (I include myself in that).
Sam