WarcraftPets has a news post that answers a lot of the general questions about the revamped Darkmoon Faire, and MMO-Champion has a great list of all the possible quests that reward tickets (needed to purchase the new Darkmoon Faire pets). Be sure to check both out. :)
For all 6 pets, a total of 540 Darkmoon Prize Tickets is required. Since ticket collecting seems very dynamic and probably depends on an individual's decision on how much time/energy they want to invest into saving up enough tickets, it's really tough to estimate how long it would take to acquire all 6 pets.
If the profession based quests and the "strength" quest is repeatable once per Darkmoon Faire week (in other words, once per month or a "monthly"), and you finish the max of all 6 trade-skill based quests plus complete the dailies every day (including the night the faire comes to town, before the dailies roll over), I estimate 2-3 DMF weeks before you can obtain all the pets. You could probably buy at least 2 pets each time the faire is in town.
Unfortunately it's note quite as clean cut as that, though.
Note that the "Quest - Item Drops" listed on MMO-Champion (towards the very bottom of the post) might be very rare drops. I haven't actually found any of the DMF Artifacts to turn in yet, so I'm not sure of the specifics and how they work.
Are the quests they begin repeatable? Are there definite dungeons, bgs, raids, situations, etc. in which they drop? Until more info is revealed about these items, a concrete estimation of how long it will take to save up enough tickets for the pets is a little up in the air. 5 to 15 tickets could drastically reduce the time in which one can buy all 6 pets, especially if there isn't a limit on collecting 5 to 15 tickets from turning in the artifacts.
So until we find out more, my best guess for pet acquisition is 3 DMFs or 3 months, two pets per week the faire is in town. It definitely could take less time depending on how the DMF Artifacts work. We'll have to wait and see.
Oh, and I'm slightly confused whether or not the faire and Darkmoon Island will be accessible all the time once 4.3 is released, but I have a feeling that it will remain a one week per month event. Someone please correct me if I'm wrong (and make me an even happier pet collector!) :P
Anyway, there seems to be plenty of coverage on the DMF, and it's all pretty straight forward so I'll leave this at that for now. Don't forget that the new zone, Darkmoon Island, also has a rare drop fishing pet to offer as well!
If you have any questions about the faire or the pets, feel free to ask in the comments and I'll do my best to find out the answer. Once the dailies reset tonight, I should have enough tickets for my first pet, so expect to see a screenshot or even video of one of the DMF cuties soon. :)