My personal order, from easiest to hardest:
Dome - you get to see what pokemon they have, so if you have a Physical sweeper, a Special sweeper, and a mixed Sweeper you should be able to handle anything (I've got a nice little Kazam, Medicham, and Arcanine that usually do the trick).
Arena - I actually love this one. Works the same as the dome, if you ask me. You just don't get to see their guys first.
Factory - if you get one good poke, you're usually set. Swapping makes this area really easy if you have decent luck. I always seem to get complete CRAP like a Hariyama with the Elemental Punches, but again .. swapping is your friend.
Pike - the random chance of being healed or of going through an empty room make this one easy enough. I always seem to get Toxic'd, though. Lame.
Tower - just like the dome and Arena, but there's that one section where you might have to rely on a computer-generated partner, so if your luck is decent this is just as easy as set KO and the Dome. If your luck is like mine, you just bring 6 guys with EQ, Surf, and Rock Slide and hope for the best.
Palace - I hate this one. My guys almost never end up using their SE of STAB'd attacks, so battles take forever. They still win, but it takes forever. Oh, and if your Heracross is going against a Dodrio, don't let it stay in and hope it uses Rock Slide. Especially when Megahorn misses 3 turns in a row and you don't find out until turn 5 that Dodrio knew Drill Peck all along :\
Pyramid - the first couple of these are usually really hard unless you bring someone with AT or something like that. Opponents always seem to use their status crap on you first turn, so poison just sucks for you. I like bringing a Blissey along because of Aromatherapy. It's not that bad once they start giving you Leppa berries.