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Ironically, there's only one menu in the game that tells you how to collect Pet Points: the menu that opens when you convert them over to your character. So the only way to know how to earn Pet Points is to have already earned them :I
But yes, every 10 levels you get some, and I think it's 2x pet points right now. You're supposed to collect them on a pet and then convert them to your character so you can rebirth pets into new/upgraded benefits.
Ironically the part of the screen that shows the pouches is cut off in your picture, as it should be at the bottom but the bar obscures it. Could you resize some windows and re-show us the image, if only to understand if the game has you marked down as supposed to having the pouches?
The most powerful skill under a bards toolkit in the meta(tm) right now is Battlefield Overture, which increases a player's maximum damage.
That said, going anything that doesn't benefit from this will put you at a major disadvantage late game. As bards build their music buffs up, you'll only be hurting your damage if you go something like Magic. Magic Attack does not get buffed by Battlefield, and Vivace only buffs the speed (arguably less so than a reforged weapon anyways).
Puppetry is difficult to use, and the puppet cannot really get enough stats to live for longer than a second in most late game dungeons. Unlike pets, which can be buffed by Divine Link in many ways, puppets lack the ability to be revived through normal means and their potion poisoning cannot be healed except at repair stations outside of dungeons. The talent's benefits are also bugged, meaning the cool-down that skills should receive from Control Marionette are not properly applied.
Battle Alchemy has the same problem that Magic Attack has.
Chain is viable, but it rewards late game players more because it depends on two stats.
Honestly for the time being Close Combat is probably the best option.
You're just going to have to run the dungeon a lot, preferably with close friends who'll give you the book if they get it. That, or spend whatever the going price is. I can at least tell you I've had it drop before, so you're more likely to get the book than you think.