I don't use braid anymore but it is useful stuff.
Personally I suggest a hot air station which we've discussed here a few times before, including the techniques of using it. I find it makes the job SO much quicker, easier and cleaner that it's well worth the small investment. I use an Aoyue 852A which cost me about the same as a single job.
As to the actual question, I'd start with the pump and clean up with the braid if you need to. A tiny bit of solder soaks up into a lot of braid so you'll end up using lots if you use it off the bat.
I strongly recommend using old mobos for practice and removing as many components as possible as practice before doing a customer jack. You'll learn the timing, the feel for old solder, the heat-soak effect of metal components etc.