Originally, prior to Cataclysm, you had to talk to a specific trainer. Speak with a Hunter trainer and he will direct you where to go.
After Cataclysm, hunters automatically get a pet at level 1.
lvl 10.
personally you'll probably be in durotar when you hit lvl 10 and i think that a boar is a nice starting pet...it has a decent amount of defense along with doing decent damage.
raptor
If you are a hunter and are wanting a new pet to level with you would have to get to lvl 10 go to your trainer then go tame a beast. If you are talking about a non combat companion you would have to go to the Auction House and go buy one or you could go to Wowhead.com and search for a companion pet.
60
1 Cynder lvl 10 2 Trigger Happy lvl 10 3 Drill Sargeant lvl 10 4 Whirldwind lvl 10 5 Drobot lvl 10 6 Spyro lvl10 7 Zap lvl 10 8 Wrecking Ball lvl 10 9 Stealth Elf lvl 10 10 Flameslinger lvl 10
Quest and do your lvl instances
A DOG
yes
Wurmple > lvl 7 > Cascoon > lvl 10 > DustoxorWurmple > lvl 7 > Silcoon > lvl 10 > Beautifly
Lvl. 10 fire and lvl. 10 ice
By completing quests and killing monsters, you get exp which goes towards your level. Vanilla WoW is lvl 60 max | Burning crusade is lvl 70 max and wotlk is lvl 80 max