You should upgrade the ability tree first as you stated, but afterwards you should, at minimum complete the basic tree (First four nodes) for attack, defense, and moves (Leave energy alone. There is no reason to touch it with so many characters to give points to). From there it's mostly up to you which tree you fill out first. Personally, I prefer to fill out attack, defense, move in that order since base stats are usually good enough to get you through most fights.
What I will mention is that for silver and bronze prize fights, a lot of the fights can be won with just the ability tree filled out, so once you figure out which characters you want to max out, it makes it easier to leave the rest with just the ability tree.
As for gold characters, think about which ones you want to focus on and save up points for them since filling out their trees can be annoyingly long due to their costs, but still follow the formula.
I can't think of a "good" character where you would do anything other than max out the talent tree first since the ability is a big reason why many of the top tier characters exist.
If you have any other questions or if I missed something, let me know.