The Beast is a true oddball class. Many of the skills level strangely, forcing you to decide carefully where you spend your skill points.

I've taken to calling it Beast even though it's called Pet in the Japanese version, simply because it seems the easiest translation that has a unique initial letter.

I'm not 100% certain on exactly what kind of attacks the various counters will deal with - I've tested a few, but it's by no means exhaustive yet. Be warned.

Same disclaimers apply, although I'll note that I liked a couple of Zaraf's translations for this job, so some of those are in here too.

Stat Boosts/Escape Up
Not in this thread.

Loyalty
Max Level: 10. No Requirements.
This passive skill will cause the Beast to take damage meant for other party members a certain percentage of the time. Note that the damage taken is based on the party member's stats and buffs, not the Beast's. As such, this isn't as useful a skill as it could be, since the Beast will take whatever damage the character it guards would've taken, modified by a small percentage based on Loyalty's skill level. Unfortunately, this skill is a prerequisite to several others and cannot be disabled. High HP is recommended before focusing on this skill. Loyalty will work for area damage and enemy skills as well, which can lead to a very quick death if you're not careful.
L1 Stats: 30% chance to guard, 105% Dmg Taken
L5 Stats: 40% chance to guard, 93% Dmg Taken
L10 Stats: 75% chance to guard, 78% Dmg Taken

Self-Healing
Max Level: 10. Reqs: L3 HP Boost.
At the end of battle, your Beast is automatically healed by a percentage of its Max HP.
L1 Stats: Heal 1% HP at end of battle
L5 Stats: Heal 5% HP at end of battle
L10 Stats: Heal 10% HP at end of battle

Self-Recovery
Max Level: 5. Reqs: L5 Self-Healing.
At the end of each turn, there is a straight percentage chance the Beast will recover from any abnormal status.
L1 Stats: 10% chance per turn to recover from status
L5 Stats: 50% chance per turn to recover from status

En Garde
Max Level: 10. Reqs: L10 VIT Boost.
One of many skills the Beast borrows from the Protector. When it activates, it reduces the damage taken by half. Enemy skills and elemental damage can be reduced by this skill. In the Beast's case, its version of the skill starts off weaker, but catches up by L6 and is far superior than the Protector's version when maxed.
L1 Stats: 10% chance to halve damage
L5 Stats: 20% chance to halve damage
L10 Stats: 55% chance to halve damage

Final Struggle
Max Level: 5. Reqs: L5 Self-Recovery.
Basically the same effect as Protector's Aegis: you have a chance of reviving with 1 HP each time you are killed. Unlike Aegis however, Last Struggle can revive you multiple times per battle. (Note: I haven't tested against Death status yet, but it's likely that it still won't protect against that.)
L1 Stats: 10% chance of reviving
L5 Stats: 30% chance of reviving 

Animal Punch
Max Level: 10. No Requirements.
This basic Blunt attack seems pretty weak, but its true power takes form as the Beast loses HP. From full HP to 76% HP, you have an innate multiplier to Animal Punch's damage of 127%. Below 76% HP, this increases to 147% until 51% HP, 170% until 26% HP, and then 196% below that. Accuracy and damage is pretty decent at high levels, and the attack is cheap TP-wise.
L1 Stats: 1 TP, 101% Dmg (128%-197%), 10% Spd, 95% Acc
L5 Stats: 5 TP, 111% Dmg (140%-217%), 60% Spd, 105% Acc
L10 Stats: 10 TP, 146% Dmg (185%-286%), 235% Spd, 140% Acc

Feast
Max Level: 5. Reqs: L5 Animal Punch.
A Blunt attack (despite the animation) that heals the Beast for a percentage of damage dealt. Reasonable TP cost, and strong for what it does. The Drain rate might not compare to other skills of this type, but the damage you deal tends to make up for it.
L1 Stats: 3 TP, 130% Dmg, 70% Spd, 95% Acc, Heal for 20% of damage done
L5 Stats: 11 TP, 202% Dmg, 106% Spd, 131% Acc, Heal for 56% of damage done

Bodyblow
Max Level: 10. No Requirements.
A Blunt attack that starts off stupidly. When you first try this skill, you might wonder what the hell you were thinking buying it. However, the stats when maxed are well worthwhile - it truly picks up after about L6, finally becoming a worthwhile and powerful attack.
L1 Stats: 8 TP, 30% Dmg, 1% Spd, 75% Acc, 1% chance to Stun
L5 Stats: 12 TP, 80% Dmg, 41% Spd, 85% Acc, 11% chance to Stun
L10 Stats: 17 TP, 255% Dmg, 181% Spd, 120% Acc, 46% chance to Stun

Rampage
Max Level: 10. Reqs: L5 Bodyblow, L10 Loyalty.
It's Multihit again, this time with Blunt damage. And it's a sickeningly powerful version, with two major drawbacks: it costs a bomb, and it's terribly inaccurate. On top of that, you require L10 Loyalty which isn't the safest passive skill to max. Still, it does have the potential for insane damage, and it can be more useful with accuracy boosting buffs.
L1 Stats: 8 TP, 70% Dmg, 10% Spd, 30% Acc, 70%/30% chance of 3/4 hits
L5 Stats: 16 TP, 90% Dmg, 30% Spd, 40% Acc, 30%/60%/10% chance of 3/4/5 hits
L10 Stats: 26 TP, 160% Dmg, 100% Spd, 75% Acc, 10%/45%/25%/20% chance of 3/4/5/6 hits

Scratch
Max Level: 10. No Requirements.
A Slash attack with a difference. At low levels, it deals low damage for a large amount of TP, but has high accuracy and speed. At high levels, it deals high damage for a small amount of TP, but has low accuracy and speed. Scratch is also a requirement for Beast Dance, which complicates things. This is one ability where you'll have to think hard about when to stop putting points into it.
L1 Stats: 15 TP, 120% Dmg, 500% Spd, 120% Acc
L5 Stats: 11 TP, 140% Dmg, 121% Spd, 104% Acc
L10 Stats: 6 TP, 210% Dmg, 21% Spd, 84% Acc

Beast Dance
Max Level: 5. Reqs: L5 Scratch.
A simple Slash-element area attack. Very nice damage for an area, only suffering from average accuracy and non-existent speed.
Area attack. Deals Slash damage.
L1 Stats: 15 TP, 150% Dmg, 1% Spd, 85% Acc
L5 Stats: 23 TP, 222% Dmg, 1% Spd, 105% Acc 

Self-Preservation
Max Level: 5. Reqs: L3 Loyalty.
A powerful and very cheap 5-turn defensive self-buff, reducing damage from all six elements even more than Fortify does (which only handles the physical elements). The drawback is that the higher levels reduce the Beast's damage output as well, but if you're using this with Feint, it can be very effective.
L1 Stats: 2 TP, 50% Spd, 100% Dmg Dealt, 70% Dmg Taken
L5 Stats: 6 TP, 150% Spd, 80% Dmg Dealt, 44% Dmg Taken

Feint
Max Level: 10. Reqs: L2 Loyalty.
Feint is simply a weaker version of the Protector's Provoke. The TP cost, Speed and Provoke rate is the same, but the gained resistance to Slash/Blunt/Pierce is lower. The Provoke effect maxes at L7: further levels will reduce TP cost and increase speed and damage reduction. At 100%, all attacks that can be provoked should always target the provoker. Feint lasts for 5 turns, and the provoke status takes affect at the start of each turn you have the buff.
L1 Stats: 3 TP, 100% Spd, 20% Provoke, 100% Dmg Taken
L5 Stats: 7 TP, 130% Spd, 70% Provoke, 92% Dmg Taken
L10 Stats: 6 TP, 235% Spd, 100% Provoke, 82% Dmg Taken

Wild Guard
Max Level: 5. No Requirements.
Essentially Protector's Parry: it will guard against any melee attack that targets the Beast during the coming turn. The base stats are identical to Parry, but it can handle all six elements, not just the physical ones - however, it can still only deal with melee attacks, not enemy skills. (All that said, I'm not sure there *are* any melee attacks that use Fire, Ice or Volt, but just to be on the safe side....) Remember that each successful guard will reduce the chances of the next one in the same turn.
L1 Stats: 3 TP, 100% base chance to parry, -40% chance per hit
L5 Stats: 7 TP, 100% base chance to parry, -5% chance per hit

Roar
Max Level: 5. No Requirements.
A skill that attempts to inflict Terror on all enemies. The chance is terribly low until you max the skill: even at L4, you only have a base chance of 4%. LUC increases the chances as always.
L1 Stats: 10 TP, 50% Spd, 1% chance to cause Terror
L5 Stats: 14 TP, 70% Spd, 75% chance to cause Terror

Lick Wounds
Max Level: 10. No Requirements.
A simple single target heal. It starts off weak, but gains in power quickly. However, it ends up only being slightly stronger than Medic's Cure, and has no Mastery to increase its power further. Combined with the Beast's pitiful TEC, you probably won't even see a 250 HP heal from this. Best used outside of battle, if at all.
L1 Stats: 2 TP, 100% Spd, 10 Base Heal
L5 Stats: 6 TP, 120% Spd, 50 Base Heal
L10 Stats: 11 TP, 190% Spd, 190 Base Heal

Catnap
Max Level: 5. Reqs: L1 Loyalty.
A healing skill that targets the Beast only. It's fairly decent, healing as much as Medic's Cure II (but without Mastery Skill), and very cheap as well. However, using it inflicts Sleep upon the Beast. No amount of resistance can prevent this infliction, but status recovery is still helped by buffs and Self-Recovery.
L1 Stats: 1 TP, 200% Spd, 100 Base Heal
L5 Stats: 1 TP, 250% Spd, 300 Base Heal 

Call of the Wild
Max Level: 5. No Requirements.
Attracts FOEs to the place you used this skill. Only works on FOEs within a certain range, and only keeps them there for a certain length of time. Some FOEs are immune. Note that the turn length begins when the FOE either reaches the destination or cannot move closer to it.
L1 Stats: 15 TP, Attracts for 4 Turns, 3 Tile Range
L5 Stats: 7 TP, Attracts for 8 Turns, 5 Tile Range

Fierce Bark
Max Level: 5. No Requirements.
Stops FOEs in their tracks within a certain range for a certain length of time. Some FOEs are immune. Extra levels increase range and duration, as well as lower TP cost.
L1 Stats: 15 TP, 4 Turn Stun, 3 Tile Range
L5 Stats: 7 TP, 8 Turn Stun, 5 Tile Range

Wild Sense
Max Level: 5. Reqs: L5 LUK Boost, L5 TEC Boost.
Chop, Mine and Take all combined into one skill with no drawbacks. Of course, you have to unlock the skill first, so maxing this takes just as many skill points as it would to buy all three for your Survivalist - but more LUC never hurt anyone.

Force Skill: Ultimate Wound Licking (Doesn't exactly roll off the tongue in English, sigh)
Party-wide heal that revives dead characters and removes statuses. It doesn't touch Binds, and it's limited in the amount of HP it can heal, but it's good in a pinch.
Stats: 200% Spd, 500 Base Heal, Removes all Statuses (includes Death, does not include Binds or Stun)