Character:
Pretty much any character that has enough speed to get to the animals and back quickly. The attributes bonuses on the relevant skills are minimal, since you would need to have high psych AND intelligence for it to matter. I would suggest to put all your points into the physical attributes and the leftover (if any) into either INT(makes it harder to steal your pet receipts) or psych (some magical resistance if you wanna use that character in PvP).

Skills:
First off you may want to decide if you want your character solely focused on taming or if you consider it more like a side profession.

The skills any tamer needs are:
Taming (duh)
One of the skills that is checked when you tame a creature. The higher your skills, the better and "older" animals you will be able to tame.
Creature Control(CC)
This will increase the number of your petpoints, which are explained below.
Zoology (Mammalia, Reptilia, Aves)
This will increase your chance to successfully tame an animal AND decrease the number of petpoints the pet uses.

To give you more options, stronger pets etc. you will need:
Advanced Creature Control(adv. CC)
More petpoints.
Herding
This reduces the penalty on your petpoints, which occurs once you have multiple pets.
Animal Care
This skill reduces the time it takes an animal to advance a level.

How does the petsystem work:
Every creature in game has a level from 1 to 100, which reflects how strong it is. The increase of HP and attack damage is fairly linear as well as the PP you need to control said creature.
Depending on the level of the creature your attempt to tame will fail or succeed. Basically your taming + zoology skill is checked against the type and level of the animal and if your character is good enough it is yours to command.

Sometimes you will receive the message: "You don`t have enough PP to control xy" This means your total PP pool is too low to fit in the PP needed to control your target. F.E. You have 50 PP and want to tame a level 100 wolf which need 68 PP the message will appear.
Your PP pool is static, IF you only have one pet! That means there are some creature you won`t be able to control, unless you increase your CC/adv. CC. Another route would be to maximize your zoology skill which effectively decreases the amount of PP a animal consumes.
Now if you want to control multiple pets every pet after the first will DECREASE your total PP pool. The skill "herding" lowers this penalty, basically enabling you to have more pets of higher levels.

Now I talked about levels, but may wonder how to get really high level pets? Well you cannot tame them, it will fail even with maxed skills. So what you have to do is tame a low level pet and let it grow to the appropriate level. In the pet window there is a bottom to enable the pet to advance and it has to be a "plus" for it to work. More on that in the hunger/loyalty section.

Hunger/loyalty
Apart from HP/attack damage/etc. every creature has a hunger and loyalty. Your aim is to have the hunger low and the loyalty high. To do that feed it regularly, once the hunger is below 400 it will gain loyalty slowly over time. But be careful: Every time you give a command, this will cost loyalty points to the point where your buddy runs away. The same happens when you hit it. If you want your pet to grow you have to keep the loyalty ABOVE 800, then it will gain a level every half hour.(less depending on your animal care skill)

The most effective way to decrease hunger is to feed food the pet likes, which means plants (barley, basileus) for horse etc. and meat/bones/carcass for the meateaters. You will notice that each unit of food decreases the hunger for a different amount. Watch out, because if you drag a stack of food into the petwindow it will remove a standard amount (about 100 I think) even if that is way to much. Try splitting the stack into 10s or even 5s and watch how the hunger goes down.
If the food you give your pet is poisonous or has healing properties this affects the animals.
I haven`t tested it, but I heard that you can feed a creature healing potions which will heal it. Someone please confirm that, would be great! If you aren`t bathing in expensive booze just wait till the pets heals up himself or use magic.

Creature Control Levels

Rabbit
Creature Control: 1-15
Taming: 1-15

Pig
Creature Control: 5-20
Taming: 5-20

Weasel
Creature Control: 10-20
Taming: 10-20

Hunter Lizard
Creature Control: 20-40
Taming: 20-40

Springbok
Creature Control: 25-40
Taming: 25-40

Urial
Creature Control: 30-45
Taming: 30-45

Bush Pig
Creature Control: 35-50
Taming: 35-50

Taur Dog
Creature Control: 40-80
Taming: 40-80

Water Lizard
Creature Control: 40-60
Taming: 40-60

Wisent
Creature Control: 45-65
Taming: 45-65

Black Bear
Creature Control: 50-75
Taming: 50-75

Chelus
Creature Control: 50-75
Taming: 50-75

Lynx
Creature Control: 50-75
Taming: 50-75

Horn Tail
Creature Control: 50-75
Taming: 50-75

Snapping Turtle
Creature Control: 50-75
Taming: 50-75

Cougar
Creature Control: 60-85
Taming: 60-85

Gaur
Creature Control: 60-80
Taming: 60-80

Razorback
Creature Control: 60-70
Taming: 60-70

Wolf
Creature Control: 60-80
Taming: 60-80

Brown Bear
Creature Control: 65-85
Taming: 65-85

Belbus
Creature Control: 70-100
Taming: 70-90

River Prower
Creature Control: 70-85
Taming: 70-85

Terror Bird
Creature Control: 70-100
Taming: 70-100

Campodon
Creature Control: 75-100
Taming: 75-95

Dire Wolf
Creature Control: 75-100
Taming: 75-100

Shore Prower
Creature Control: 75-?
Taming: 75-90