I have a feeling the OP was asking which kinds of villagers to get in their town for a pet shop theme, not saying they want an in-game pet shop, but I could be misreading it
Anyway, I think a good idea would be to have some combination of the following species: birds, cats, dogs, hamsters, and rabbits. Those are all commonly kept as pets. And then a nice idea would be to have them only "natural" colors, for example you can't really find a purple cat or a pink dog in the real world! Some good villagers for this would be:
Cole, Dotty, O'Hare, Ruby, Bonbon, Soleil, Flurry, Hamlet, Hamphrey, Walker, Shep, Portia, Goldie, Butch, Bones, Purrl, Punchy, Merry, Lolly, Kiki, Katt, Sparro, Jitters, and Robin. There are possibly some that I missed as well
To keep a balance of genders and have one of each personality--also based on my personal preference--I'd go with:
Katt
Punchy
Portia or Purrl
Hamphery or Butch
Goldie or Lolly
Jacques
Hamlet
Ruby
Jitters or Sparro
Dotty
That also potentially gives you at least 2 cats, 2 birds, 2 dogs, 2 rabbits and 2 hamsters! Of course, you can modify my list to your own preference, or throw out the whole thing if you just hated it xD But I hope that helps.