Derek has mentioned basically all of the positives of contracting - the downsides are that shorter contracts (three months, or anything less) tend to be either quite dull (very routine work that nobody else has the time for) or unpleasantly intensive (desperately trying to ship a disaster); in some organisations you will be treated as "just another bloody contractor" by the perms, who will know you make more money than they do and hate you for it; you may get less responsibility than you would have as a perm, or generally "less say"; and if you end up contracting for the sort of company that you'd hate working for as a perm, it will be just as bad as a contractor, except you'll probably care even less for them because you will know that you can quit very easily, which is, itself, a bit demotivating.
I transitioned from contracting to running my own consultancy, which is much harder work, much more stressful, and with a lot more risk, but quite a bit more rewarding (for me personally - definitely not something I'd recommend to everyone).
Emails always gladly received!
Starting a new contract is just like starting a new permanent job — hand in your one month’s notice and move on. Though, I tend to just quit, take a couple of months off, and then find the next contract, but most contractors move directly into other contracts with no down time.
(Stay well clear of Computer Futures. They have a terrible rep in Ireland and in the UK.)
Hays is a huge one and I've done one contract through them which went very well. They seem to be well organized, know when I'm becoming available and more or less what kind of work I'm looking for.
Computer Futures on the other hand spam me with tons of irrelevant stuff. That is partially my fault however, since a long time ago I submitted a CV to them with lots of keywords that I don't care about any more. Having said that, I recently got an interview through them and the client wanted to hire me. Ultimately it didn't work out, but I don't think they are to blame for that. I certainly did not put them on the blacklist. :)
Some of the other ones are smaller agencies from the UK and Ireland, where I am regularly in touch with one or more of their recruiters. You can probably find all of them on LinkedIn. The good recruiters often post new contract opportunities including rates (a nice habit in the UK, still uncommon in Germany).
They all love phone calls, which is a bit annoying, but not the end of the world, IMHO.