I'm definitely not an expert, I only started doing BD during the Wraith Resurgence event. Some of the weapons I use aren't buyable I believe, as they're from that plot.
But my main used weapons that are buyable are Yooyu Knuckle Duster and Turned Tooth. I also have Patched Magic Hat and Greater Healing Scroll but to be honest I don't use them really.
My battle pet's stats aren't super super high. She's level 74 at the moment. I recommend getting to at least level 50 because then you can train a really valuable battledome ability that completely blocks the enemy's moves for one turn (there will be guides on that). Random dailies add to her stats, and I use Faerie Quest Fortune Cookies from the NC Mall and try to make sure that she is always my active pet so she gets the rewards from the faeries which are almost always stats based. When you're training, don't worry about movement/agility. As far as I know it has no effect on results. Endurance (hit points), attack and defence will be what makes a difference. I've never battled another player. I started with Donny as I often got fairly valuable Neggs or decently rare items. Once you get your stats up, look into how to get the Koi Warrior as an opponent. That is mostly all I play now, to get as many red codestones as possible to sell. Again I'm not an expert at all but hope this helps a bit!
Oh and I use Dewberry Revivers in between battles when needed for complete healing (I collect cheap direct from Healing Springs)!
But look up which potion you need based on what HP your pet has!