I use both, it just depends on the situation.
A desoldering pump usually does a better job removing solder from a plated through-hole but isn't quite as effective on SMD pads.
I'll second the Engineer SS-02. It's the best manual desoldering pump I've used in 20+ years.
As for wick, a roll of Chemtronics 10-25L Chem-Wik will last quite a while and is the best I've ever used. It's already coated with rosin flux so it sucks off solder like a magnet.