~~Everything in this post is coming from someone that has not done any Seekers content yet, so keep that in mind.~~
Capping Reward's potency is one thing. These days, you can cap the 50% with two pieces; Totemic Jackcoat (Reforged AF body) for 20%, and either Monster Gaiters +2 (AF2+2) or Ankusa Gaiters (Reforged AF2) for the other 30%. Other slots can be filled with MND and Reward Recast.
Also, Monster Gloves +2/Ankusa Gloves and the merit trait Beast Affinity allow you to raise your pets level.
For merits, definitely Call Beast and Reward. For group 2, Beast Affinity 5/5 and Feral Howl 4/5 with the last point into Beast Healer. For the longest time, I figured it was useless, but that little bit of extra Regen added to Reward has been pretty helpful.
As far as gear goes, I'm still working on that myself. I recently came back after being one a year, and haven't done any Seekers content yet. However, Reforged and Sparks gear has been doing me wonders so far. Most definitely look toward getting Regimen Mittens from AA MR. Anwig Salade from MKE with 10% DT is also an excellent piece. The other slot for a buff can be either Pet Haste (by far, the most popular choice), or Pet Regen. Again, that tiny bit of Regen on a pet with 3k+ HP may not seem like alot, but it has definitely helped me. Particularly when coupled with the DT. Ferine Necklace and Sabong Earring are nice as well, as each adds 2% Pet Double Attack. Another thing that is a big boost is Skadi's Jambreaux (and the +1) from Salvage. Unless something new was added with Seekers, Jambreaux are BST's only Movement+ option.
For axes, I would say the ultimate goal for any BST would be Aymur, but as that is a pipe dream for many, there are definitely some worthwhile pieces for us mere mortals to attain. First and foremost are a pair of Pet PDT Magian axes. 11% PDT x2 is most certainly a huge boon. The axe from AA MR is probably the big prize most BSTs are after right now. Others that offer utility include Glyph Axe from the Imp NM Bakka in Abyssea-Konschtat, Aalak' Axe (I got one from AH pretty cheap) that has pretty good melee stats and Pet Double Attack, and of course Eminent Axe.
There are also a number of belts that are good for pets now. The first that comes to mind is Moepapa Stone, with its 5% Haste for both BST and pet. Another is Kuku Stone with 6% Haste for the BST and 4% for the pet.
I'm certainly not the brightest shining example of a great BST, but I did manage to build my Gjallarhorn solo in Dynamis. ^^