To add on to the above post:
Attentive Ibuki - Absolute best all around pet available at the moment. Extremely sturdy, only a few monster-used physical attacks in the game do not count as Slashing damage (this includes things that seem obvious, like ranged attacks), meaning that Ibuki has a natural 50% extra resistance to almost every type of physical damage above any other pet. In addition to being extremely survivable, he comes with some remarkable Ready moves. Molting Plumage is consistent, decent magical damage and Swooping Frenzy is less powerful, less consistent physical damage with an additional 25% DEF down and -25 MDB attached. I use this to solo Easy and Normal AA fights. Only downside to this guy is the cost, but you save a lot by making the jugs yourself if you can.
Caring Kiyomaro - Very cheap, and very effective all-around pet. Doesn't have any special damage resistances besides a pretty high resistance to ice damage (25%), but his basic melee attacks and Ready moves are pretty potent, and he has a natural chance to Counter (he's effectively a Footwork MNK) and nicely sized HP pool. Sweeping Gouge deals pretty decent physical damage in addition to lowering the target's DEF by 25% (was pretty surprised to find this when I first tested it). Zealous Snort is neat too, giving him a nice boost to defensive stats (and, less importantly, haste). Very nice pet to use for most situations where you don't need AoE damage or a very sturdy tank.
Lucky Lulush - Looking passed his lower level cap, Lulush is probably my most favorite pet. He's extremely cheap and has a great Ready move pool. Wild Carrot helps reduce the cost of pet food and is useful as an emergency heal if you get into a bind, Foot Kick is a very strong attack, matching most other Ready moves in fTP (1 -> 2 -> 3) but forcing a critical hit in addition to getting a natural 6% critical hit damage bonus. The usefulness of Foot Kick is particularly apparent with Aymur, but since that's so uncommon it's just a bonus. Whirl Claws is what sets Lulush apart from other pets the most: a very consistent source of AoE physical damage. With Unleash, you can solo things like Einherjar TIII wings using this little guy.
Droopy Dortwin - Until he's cheaper he isn't very useful, but he'll basically be a stronger Lucky Lulush once they make the mushrooms drop from funguars.
Faithful Falcor - Useful until you get a Chaac Belt. Comes with a nice set of Ready moves in addition to natively high Evasion and a TH1 trait. Best used in Dynamis for the TH, but Choke Breath also serves as a means of dealing non-elemental, physical and magical resistance bypassing damage (as well as a low cost, low risk extra JA proc for dynamis). Back Heel is effectively Foot Kick without the autocrit or additional crit damage attached.
The rest really aren't very good. Gorefang Hobs can be the best DD pet we have in certain conditions since he gets a pretty big attack bonus, but it isn't often that you're using a pet solely for single target damage output. Malfik, Xerin, Henry, and Hermes are all pretty weak in general and are hard to argue a case for.