If you have a low weapon skill but just got a brand new shiny weapon, where do you go to level it?
I go buy as low a level as possible weapon of the same kind, with as fast attack speed as possible, and head straight to Gnomeregan. From there I hop right down into The Hall of Gears where the Corrosive Lurkers place a stacking debuff on you that reduces your damage dealt. You can afk auto-attack for quite some time like this!
Where do you guys go?
Just level a gathering profession and kill everything as you go. Make money while you skill up.
I run around in Elwynn killing lowbie Defias and cows.
No offense 
find mobs and hit them, really don’t matter where.
I mean, obviously a little lower than your current level is a good idea. unless you have some heals with you
I would usually just fight green level mobs. So long as I can kill 1, everything is fine. It’ll go up quickly, 5-10 mobs and it’s high enough to fight normal enemies again.
Turtles. they don’t do a lot of damage so you can sit there and hit them for over a minute without even healing and not die.
It depends. What level are you? Are you 54, or 14? It makes a big difference in where you go.
at 60 on this guy I would go to the unkillable mobs in Blasted Lands, use seal of light and do it that way,
Blasted Land Servants
Nath’d as I was typing. 
This is the way to do it.
it worked for a few , got my unarmed maxed that way too :O)
I always load up on Int gear and go to a lowbie dungeon, normally stockades because i’m lazy and it’s right in the city.
I just fight whatever is in front of me that I can wail on for a while without dying…
I tend to spam Hamstring because it is fast and counts without waiting on weapon swing timer (just rage). I usually look at quests I’m on for something green where I need to kill green/gray mobs, and know it will start slow while I get the early points.
Wherever im currently at. Let my pet tank and faceroll R1 Wing Clip. 
I show up to the dungeon with no weapon skill and let it level up on its own.