There is a wiki guide to the tournament http://www.wowwiki.com/Argent_Tournament
The wiki is nice because it actually has a nicely annotated map of the tournament grounds.
In Dalaran there is a representative standing near the flight master who will give you your first quest to head up there. This will start the chain. There is the main tent to the west that will give you a quest to head to your factions' tent. Regarding pet collection:
You can only work on one race at a time. (race = faction-city)
You will need to collect 25 Aspirant seals in order to complete the Aspirant Challenge quest. If you do all the daily's you get 5. So it will take you 5 days to be able to complete the aspirant challenge quest. Once you've completed this and become a Champion of whatever race you've chosen then a vendor opens up and you can purchase various items including a race-specific pet.
You don't need to reach exalted with each race in order to buy the pet, only revered.
Each pet costs 40 Champions seals. How do you get Champions Seals? I'm glad you asked.
You get Champions Seals by completing another set of daily quest that are given to you in the main tent, but you'll have to work your way through a quest chain and I believe you have to have unlocked at least Champion status with one race. So if you don't see those quest available right away, keep plugging away. Once you've reached champion with one faction-city, you should be able to start earning Champion Seals.
When you do start to do the Champion Seals quest in the main tent - you get to an option of quest rewards, either a bag of gold or a Champion's writ which you can use to purchase commendation badges and these increases your reputation with the various races. One thing that it doesn't tell you is that if you chose the bag of gold, then you get an extra Champion Seal.
Each faction-city has their own pet, so if you're going to want to get them all you're going to need to do them on a horde toon and another on an alliance toon.
If all of this seems daunting, they are tradeable, so I see them selling in the AH for anywhere from 500g to 2500g.