At the bare minimum it feels like the most important thing you need to do is collect your dog ingots. Every time I go back though I compulsively do everything .
One thing I think isn't worth doing is fishing, unless there's some way to guarantee you get the rare fish every time the bond fragment reward is low while it takes a decent amount of time to do all three.
Petting and feeding Sommie plus sweeping his cave is a relatively quick way to get bond fragments. Cooking and working out are obviously nice for the stat boosts, but I could pass on them. Wyvern ride is looooong. Arena is quick, but I could also pass on it.
What do you all usually do? I feel like I have a huge amount of bond fragments, but I haven't been making or melding many bond rings which seems to be the biggest sink for them.