What kind of load do you expect to carry? A pack full of photo gear will call for a sturdy set of boots while with a single camera with all-purpose zoom you can get away with a pair of light, low walking shoes. Personally, for the varied environments you've listed, I'd go with a different set of shoes for each of them. You don't need "waterproof" (personally, if I'm worried about wet, I'll go for GoreTex linings) for the desert, but you do (probably) want something to keep sand out. And for mountains (I'm assuming rugged/rocky terrain) you'll want good support up the ankles (as would also be the case for heavy gear loads). For relatively smooth woodland trails (not heavy with roots or rocks across the trail), walking shoes would probably do well.
As for brands, every company uses their own last - one that works for me wouldn't necessarily work for you. Find a good outdoor supply store (REI, EMS, LL Bean) and try on a couple of pairs. Go in wearing a pair of snug-fitting, but comfortable, shoes; not slip-ons or chukka boots. You want your feet to be used to being supported all around when you start trying the new shoes on. If one pair feels reasonable at first, walk around in-store a lot. But don't try too many pairs the same session - your feet won't be able to tell you if they hurt from this new pair or from the last pair you tried.