There are two possible answers that immediately spring to mind:
1: Focusing on PvP
In focusing on PvP to level up, spiders should be a fairly uncommon sight, although they do seem to be popular pets for hunters.
I've never personally leveled up a character using solely PvP, so I'm not entirely sure how the speed of it relates to the usual speed of leveling up, but I do believe it compares at least comparably.
2: Focusing on random dungeons
There are, at least insofar as I can recall, no spider-oriented dungeons, and pulling random dungeons on repeat is a good way to earn XP, as being in a random group increases XP gained.
Again, hunter pets might be a problem, but to the best of my recollection, they are more of a PvP-pet, so there should be limits to how often they are encountered.
Edit: As mentioned in the comments, there are a few dungeons with spider-bosses, and I guess you could classify Azjol-nerub as somewhat spider-oriented.
In the general case, however, most dungeons tend to stick with a theme wrt. to NPCs that inhabit them, raid dungeons excepted, and most are pretty humanoid-heavy.
If you want a more quest-driven leveling up while seeing a lot of Azeroth, it's probably easier enumerating areas you don't want to go into:
Spiders feature prominently in the climes where one would legitimately expect them to turn up:
Dark and scary forests.
Duskwood is among the worst offenders, but there's a fair few spiders in the Plaguelands as well.
Dustwallow Marsh is also rather spider-ridden, and parts of Stonetalon Mountains are as well.
For a Horde character, the progression through The Barrens, and simply going south through Thousand Needles, Tanaris, and Un'Goro Crater should provide a fairly Spider-free progression up to about level 50 or so. After that, Winterspring should provide a suitably spider-free existence leading to level 60.
Once you get to 60, however, your choices are somewhat limited, but fortunately, there is little in the way of spiders in Hellfire Peninsula, after which a normal progression through the Burning Crusade content, avoiding Terrokar Forest in particular, should give a quick level up to 68-70, at which point you can go to Northrend.
Starting in Howling Fjord and working your way through Dragonblight and Grizzly Hills should pave the way for a quick jaunt through Scholazar Basin, which is basically Un'Goro 2.0, and largely (entirely?) spider-free, focusing on big game.
Going then into the Cataclysm content, I can't really remember any particular spider infestations, but where there are undead, there are generally nerubians.
Still, Uldum and Vashj'ir are probably the least spideriffic of the lot.
On a completely unrelated note, and as has been pointed out in the comments, perhaps using PvP and framing the enemy hunters and their pets in an immersion therapy type thing could be helpful going forwards, but that' s really neither here nor there, I suppose.