To start you need:

Tamer Job (Kit)
Cooking Job (Trixie)

Farmer job, I think you start with it now?

The pet skills are really handy, so here's how to level fast and get to the good ones.

First, grow ALL Sunflowers. (Buy seeds from Gary) Then buy the Bird Seed recipe from Trixie, make sure you're making bird seed from the flowers constantly. This helps level quickly when your petting runs become slow.

Don't forget to pick up a Fluff Brush from Gary. You may want to continue petting even after you're EXP drops for the fluff.

Visit these lands, its either once every hour or once every 6 hours I'm not sure because I have before work, at lunch, and after work I can get on and actually work on my skills instead of just pop in real quick to jump in a chair or something. These may change at any time since the owners can move or add more pets but here's what they had last time I counted.
These numbers are now likely to be very wrong since the change to doghouses people are keeping much less chickens.

Turkeys and Bunnies I'm counting as "chickens" because they too have birdseed for their favorite food.

13 Pets - 0 Chickens

59 Pets - 57 Chickens
46 Pets - 39 Chickens - Also 50 pots they seem to keep full if you want farm exp.
39 Pets - 27 Chickens
25 Pets - 17 Chickens
38 Pets - 14 Chickens
31 Pets - 12 Chickens
24 Pets - 11 Chickens
22 Pets - 10 Chickens
56 Pets - 9 Chickens
15 Pets - 8 Chickens
15 Pets - 6 Chickens
12 Pets - 3 Chickens
11 Pets - 2 Chickens

If you are under level 15, DO NOT FEED ONLY PET. That will net you 391 EXP (amount may change at any time as people change their lands) each time you run these farms. I'll add more farms if I find more.

Once you are level 15, your petting exp lowers, Now use all those bird seeds you've been making and hit up peoples chickens. I've noticed I'm not the only one that's figured this out since chickens are often fed while other pets are going hungry. Chickens favorite food is bird seed (obviously) but the important part is that you get DOUBLE exp for feeding the chicken the bird seed. So even if you only visit the two most chicken heavy farms on the list, its 96 chickens, you'll get 192 exp if you only feed each chicken once. If you're lucky and no one else has hit them yet you may be able to feed the chickens up to 5 times each for 960 exp on just the 2 farms.

But in order to keep that up you need to have kept up on your sunflowers/birdseed in the first 15 levels or you'll run out fast.

More explanation about how feeding exp works.

Kogo now count for Tamer EXP too!:

While in your petting phase, remember to look for kogo on peoples land as well. If you click their kogo and throw a ball it counts for exp at the same rate as petting. It also cuts off at 15.
Capturing Kogo will net you exp seemingly depending on the rarity of the kogo.

If you want to hide your pets:

If you want to keep your pets feeding points to yourself, they're your pets its perfectly okay if you want to do that. Two ways to do it that I know of. First you can place the pet on your land, feed it, then pick it up and leave it in your inventory until its hungry again. You'll have to place it back down on your land so see if its hungry or not. This way you don't have to arrange your whole land around hiding the pets, but if its a chicken or some other product producing pet they don't produce anything without being on a doghouse. The second is to buy a lot of cheap land items like those Christmas candy cane polls or flower pots or anything you can build a lot of. Place your doghouses along a top side or corner, then use the items to fence off that area. The avatar needs a 2 block (I think) hole to walk through so by placing a small item every-other square people can't go through and get to them. When you want to feed your stuff just pick up one of the pots or whatever item you used and go through yourself. Just remember to place the item back down before you leave or you'll leave it open to scavengers again. This way you can still get Eggs/Milk/Wool off those pets. DO NOT stick your pets right up against the pot line, people will still be able to reach them. Back them up a little bit.

Thanks for listing those farms! I didn't know about the exp drop off for petting, that's really good to know

I hope that they add more items that can be fed to other animals that gives double xp. It would make the grind less of an issue.

if you are dedicated to it. It isnt that hard honestly. I have on my way to level 29.

I should clarify the bird seed and double exp.

Its not exactly double exp, its double feed which in turn means you can get double exp, not guaranteed to get double exp.
You can gain fractions of exp, this is how your petting exp lowers instead of getting 1 exp per petting you get .1 exp per petting. The same works for feeding. You get .1 exp for every 1 point of feeding you give the pet.

1 Pet food is worth 10 points to the pets feeding meter so you get 1 exp per feeding. But if the pet is already at 97 hunger, and you feed it then you'll only get .3 exp because you only added 3 points to the pets hunger meter. How this effects chickens is when giving a favorite food 1 food = 20 hunger points instead of 10 hunger points. So 1 birdseed = double exp IF the chicken is at 80 hunger or less.

But if you come by and see a chicken at 85, you probably want to feed it and take the 1.5 exp or else someone else is likely to take it anyway. Trying to wait until the chicken is 80 or under leaves it open for anyone else whose willing to take slightly less exp to snatch the feed opportunity away from you. Birdseed is way cheaper then pet food anyway so unless you're struggling to keep birdseed on you might as well take the points you can get.

It also seems like the hunger is tied to the pet itself, probably to prevent cheating. But what you can do with your own chickens if you find people have been feeding them and you want the exp for yourself is feed them, then pick them up and let them sit in your inventory. Just put them down when you want to feed them so no one can grab the hunger points form you, but you can't get eggs from them that way.

Technically only 39 pets on my land that you can reach - the rest i have put out of reach so i actually get exp from feeding them XD

I figure I'll do something about the feeding thing once I hit level 15 and it matters more, but for now I definitely don't mind people keeping my chicken factory running for me ;D

I added moar chickens to mine xD I have to say that even tho I luv bunnies, the shop one scares me a bit. Bunnies are supposed to have long back feets! I want those april fools baby bunnies by the dozen back again

Also the goat is my favorite most handy pet OMG xD

I keep my Christmas Goat fed and petted. It does a really good job.

I take care of my goat too but honestly he's the last to be fed. When I do have the extra pet feed for him I love his cleanup job.

Updated some numbers since some lands changed. I also added how to hide your pets if you want to keep them to yourself and added chicken counts on the listed lands.

If you want to add my land you can, I don't mind helping people out. I have 27 chickens on mine and a few other animals.

I added you to the list!

Just to help...I've been MIA for my semester, but I've some 13 pets on my land. Anyone's welcome to visit for w.ev points they need.

Oh! Bunnies like bird seed? Wish I'd known sooner, would have loved to save the regular feed for my cow and goat pets.