Generally, always press, never saw. You want clean cuts, straight down.
Cheese knives for consumers, IMO, break into two categories:
- personal preference
- practical
There are myriad handle and blade shapes and for the most part it's personal preference. Because they are often presented on a cheese board at parties, etc. there's more reason to consider the "art" of them rather than simply utility. Artisan cheese is an "affordable luxury" so many people want a knife that reflects that will still being functional. Truth be told - I've seen too many that err on the side of art and are not fun to actually use.
The practical comes in with:
How hard is the cheese? How "sticky" is it when cutting. You need heft for the first and less surface area for the second.
Somebody gave me these and I find I use the two on the right quite often. The hard cheese knife is handy for breaking very hard cheeses. The other does a decent job with softer, stickier/gripper cheeses.

The better knife for sticky cheese is something like Global's with as little surface area as possible.

For the softest cheeses, at least for photography I actually just use a sturdy piece of fishing line.
Having a chisel/spade is handy for breaking big chunks of parm, gouda and the like:

You can use ordinary heavy knives as I did in the pic above- but if the cheese cleaves, down goes your knife tip into the slate.
On the commercial/practical side, for larger/whole cheeses, there's one cutter you could use for just about everything and that's a commercial cheese wire cutter from
Handee. Takes a little practice but once you get the hang of it it's the best way for breaking wheels, slicing quarters etc. But 99% of consumers don't need this.
Edited by Cary Grant - 10/23/12 at 11:50am