I'm an achievement nut, always looking for new tasks to do for pixel rewards so here's some I think would be cool to have. I'd like to invite anyone reading this to suggest some as well!

Full spectrum - have one pixpet of each of the 16 colours in your inventory
Favourite pet - have 5 hatched pixpets of the same species in your inventory
Favourite pixpal - have 5 hatched pixpals of the same species in your inventory
Full cookbook - fill out your recipe collection
Favourite colour - have 10 decorative items in your inventory of the same colour
Skilled fighter - get a pixpet to adventure level 10
Warrior - get a pixpet to adventure level 30
Champion - get a pixpet to adventure level 60
Veteran - get a pixpet to adventure level 100
Glitch hunter - defeat 10 glitched pets
Glitch tamer - defeat 30 glitched pets
Healer - turn 10 glitched pets into eggs
Medic - turn 30 glitched pets into eggs
Are you REALLY sure though? - Decline when Baluna offers you a biscuit 10 times with the same pet
Utterly lost - spend a full 24 in-game hours just using the "explore" option in one location
Artist - have a submission approved for the gallery
Pixcoin saver (bronze to emerald) - have 100 - 1,000 - 5,000 - 10,000 - 100,000 pixcoins onhand
Inquisitive - click the link to the wiki

EDIT: fixed the category to what I think is the right one, also added a couple and changed the proposed names of others

Thanks for the badge ideas, will see what we can do about them in time XD

Good ideas, it would be good to see some new badges!
I would probably change the names of "Beast slayer/Massacre" to something like "Glitch Hunter/Glitch Tamer"? Massacre seems like the pixpets are getting killed or turned from zombies to me.
Otherwise these all sound great to me!