BProud I was a union organiser in my last workplace for four (long) years. It was tough at times but I learned loads about negotiation and management as well as employment law. You also get to understand the guts of your organisation in a whole different way. I don't do it any more but would echo a lot of the points above. One thing I and my fellow rep found useful was to have 'office hours' for TU stuff- or you are constantly being interrupted in the middle of other work. We also found it useful not to rep processes that were with people in our immediate teams, so picked our casework accordingly.
Our area rep was pretty useless to be honest, but we were not a high profile industry so I suspect we did not get Unite's finest...
One of the most difficult things to communicate to people is that 'unfair' (as in not very nice for the individual) is not necessarily illegal- yes, it's horrible when you're made redundant, but if the process has been carried out correctly, there's not much we can do about it. Make sure you don't over-promise to people on those grounds- a lot of what you do will be about process rather than principle.
Also agree with TheFarSide on people hiding behind you- the amount of times we had members say they would be prepared to take industrial action on something at the outset, then lose all appetite for it as we went through the process was infuriating, because you'll put a lot of time and effort into negotiating with management with that as your final position. You then look a bit daft if that is not supported.
Another piece of advice would be to make sure boundaries are clear- you are not someone's counsellor or even their friend, and remain at a professional distance (personal fave was when someone's Dad emailed me about their grievance process). When people are upset about something at work they can't really talk about, they will stick to you like glue. You'll also have to deal with people who you think- 'yes absolutely- you should be on capability' which can be a bit tricky- again, that comes back to the process and keeping focus on that.
I am very glad I will never have to run a salary negotiation again, too...