Some of this will just be restating what others have said, but I'll add my own info in here as well.
Merits:
Beast Affinity 5/5
Beast Healer 5/5
Call Beast Recast 5/5
Reward Recast 5/5
This is what I have merited, but this might not suit your style of play. Some bsts who DD along with their pet might go 5/5 Killer Effects in place of Call Beast Recast, or 5/5 Killer Instinct in place of Beast Healer. The killer effects stack very nicely with the Emp +1/2 body.
Equipment:
"Pet:" Trial axes are a must, I would pick x2 of the following, or just do x2 of all 3. It also wouldn't be a bad idea to make a STR/Atk axe for yourself for Ruinator.
Pet: -pdt - These will be used in most situations and are usually made first.
Pet: Evasion - In certain situations these are better to use in place of Pet: -pdt axes. If your pet isn't getting hit it saves you reward food.
Pet: +mdb - These are super situational, however they are very easy and cheap to make. If you're fighting magic heavy mobs a lot, these can be useful if you don't want to use the slug pet.
Gylph Axe - Great for idol set, and not a bad axe to hold you over while you're make Trial axes. Drops from Bakka in Aby Konschtat.
Zoraal Ja's Axe - Can be situational depending on how you equip yourself for reward, but this is pretty much the standard. Enhances "Reward" Effect, gives +10% reward potency.
I never looked into making any of the Pet: Atk/Acc axes, pets are already strong enough as it is. I don't see them making a noticeable difference that would constitute making/using them over any of the above choices.
Moogle Add-on Helmet - Anwig Salade (w/ Pet: DT -10%, and either Pet: Haste+5, or Pet: Regen+1.) The Pet: DT -10% is a must, the other is a toss up. You can play it safe and get Haste +5% which most people do, or you can get Regen+1 and have a nice idol/tanking helm.
Spurrer Beret - Only get this if you chose Pet: Regen+1 as your 2nd augment on your Anwig Salade. This is one of the few pieces that enhance Pet: Haste. Drops from Morille Mortelle in Crawlers' Nest [S].
Stout Bonnet - There are 2 different versions of this, the Bison/Brave and the Khimera/Stout sets. They all have the same stat, Reduces "Reward" recast time, but the HQ's reduce it more. NQ reduce reward time by 10 sec, and HQ reduce it by 15 sec. It has also been debated about the effect of the Stout Bonnet, as testing has shown it to reduce reward recast by 16 sec. You be the judge.
Ferine Gausape +2 - Great bst DD body. But if you like to let your pet do all the work there is another option.
ACP Add-on Body - Mirke Wardecors (Pet: DA/Crit rate +2, and Pet: Atk +15) This is the best DD body for your pet, the DA stacks nicely with the Ferine Necklace, the added atk is just icing on the cake. You can also augment Pet: Acc and Call Beast delay -15, but those are much less useful and can be replaced.
Beast Jackcoat +1/Monster Jackcoat +1/2 - The stat Augments "Reward" removes Paralyze, Poison, Weight, Slow, Silence and Blind from pet. Don't bother with the NQs unless that is all you can get.
Oneiros Cappa - This is optional, but it stacks well with the other optional gear. Gives Pet: -pdt 3%. Drops from a NM in Dynamis Sandy.
Ferine Mantle - This is pretty much the standard back piece for bsts. It gives Pet: Acc +10 (1 of a few reasons not to get Pet: acc augments on the add-on body). Drops from Rashovnik in Aby Konschtat.
Ferine Necklace - Another standard for bsts. Gives Pet: DA +2%. Stacks nicely with Mirke Wardecors with DA/crit rate augments.
Shepherd's Chain - This is optional, but don't pass it up because it looks gimp. It has its uses. Pet: DT -2%. Drops from Okyupete in Misareaux Coast.
Monster Gloves +2 (w/ Augment) - Put those 5/5 Beast Affinity Merits to good use. Augments "Call Beast" and the augment "Enhances 'Beast Affinity' effect". These are a game changer for bst. You only need to equip them when you use Call Beast.
Beast Bazubands - Gives Pet: Defense Bonus (defense +5%). It doesn't seem like much, but they do make a difference. Drops from Dragonscaled Bugaal Ja in Mamook.
Bestia Mufflers - Random augments using Battle Trophy: 4th Echelon Key Items at Dominion Tacticians. Possible Augment of Pet: Subtle Blow +1-5. Pet subtle blow is very nice when stacked on WS heavy mobs.
Ogre Gloves - Don't bother getting the HQ, they are pointless. The stat Enhances "Reward" effect gives reward potency +10%.
Moepapa Stone - Pet: Haste +5%. That is all I need to say. Get this period. Also has Haste/CHR +5 for yourself for an added bonus. Drops from Lost Fairy Ring in Dynamis Valkurm.
Selemnus Belt - This is optional, but like the Shepherd's Chain, don't pass this up, it has its uses as well. Gives Pet: Mdt -3%. This stacked with Oneiros Cappa and Sheppard's Chain will give your Pet: 5% DT. Like I said it has its uses. Drops from Frogamander in Garlaige Citadel.
Ferine Earring - Can be useful just for the Pet: Acc+3. The Enhances "Reward" effect is useless really. (another reason not to get Pet: acc augments on the add-on body). Drops from Battlerigged Chariot in Aby Altepa.
ACP Add-on Augment Earrings - Emerald Earring gives a possible augment of Pet: Evasion +1-5. Ruby Earring gives a possible augment of Pet: Attack +1-5. Spinel Earring gives a possible augment of Pet: Accuracy+1-3. These are possible rewards after Examining the Treasure Coffer in the Tenshodo with an Ivory Key. These are amazing if you can get your hands on them. There are no other Pet: earring options besides Ferine Earring, and these can be better.(And another reason not to get Pet: acc augments on the add-on body).
Ferine Quijotes +2 - Best thing about these pants is Stout Servant +1. The rest is an added bonus.
Monster Trousers +2 (w/ augment) - Reduces "Reward" recast time by 10 sec. If you've got these, a Stout Bonnet, and 5/5 Reward Recast merits you can have a 49 sec Reward recast timer! The Augment "Enhances 'Familiar' effect" adds Pet: Haste +10% while under the effect of Familiar. Pants need to be equipped when used, but you can remove them and keep the buff!
Bestia Breeches - Random augments using Battle Trophy: 3rd Echelon Key Items at Dominion Tacticians. Possible Augment of Pet: Subtle Blow +4-8. Stacks nice with the hands.
Demonry Core - Ammo for your pet? yes plz! Gives Pet: Acc +4. (Last reason not to get Pet: acc augments on the add-on body). Drops from Marquis Caim in Dynamis Xarc.
Monster Gaiters +2 (w/ Augment) - Enhances "Reward" effect III. Get these for your reward macro. Increases Reward potency by 30%. With a Zoraal Ja's Axe, and Ogre Gloves your reward potency will be capped at 50%. The augment "Enhances 'Beast Healer' effect" is godly when stacked with 5/5 Beast Healer merits.
Armada Sollerets - Pretty much the best Pet: feet in the game. By themselves they are not much, but after you augment them they become your pets best friend. Possible augments that are worth mentioning (Pet: Haste +1-4%, Pet: Subtle Blow +1-7, Call Beast Delay -2~10).
Wow that ended up being way longer than I had planned. -.-;