I was wondering which quests there were and what to do for them.
If in doubt, check the wiki out! Here's a page of all the obtainable pets in game at the moment. (https://legendsofequestria.fandom.com/wiki/Pet)
Note that the turtle quest is for animal talent earth ponies only and the dog is obtained by answering 'It sure is. What're you going to give me?' to Astral when her quest is nearly complete.
Pets are unfortunately not tradeable, so if you miss out or accidentally delete them, you won't have the option to get them again later unless you make a new pony.