In my experience, braid works OK if you have a relatively hot and large tip on your iron. You can buy braid impregnated with flux, or just the raw copper braid. If you are new to this, you probably want the flux-impregnated braid.
One tip I recently came across (after being involved in electronics, one form or another, for 20 years!) is thus:
Have a hard work surface to hand. Apply heat to the solder for a few seconds... and when it is good and molten, quickly slip the iron to the side and BANG the board down HARD on the work surface. Provided there is a small gap between board and work surface, the solder will splodge out. This does actually work very well, a fairly clean solder removal, no flux required.
I normally do this on PCBs I designed though... so even though my career is in electronics, I would be VERY wary of doing this with anyone elses boards. G-forces can fuck up many crystals, you have been warned!
best regards,
stephen