You can't get the turtle as a pet. It's just a wild animal, hanging around. There's also a racoon that's near Vestas farm, it usually hangs around there near that road that goes into Mining Town or whatever.
The only pets you can get are of coarse, your farm animals(cows, sheep, goats, bulls, a horse, chickens, ducks..). And your dog. I think you can get a cat too, but, I'm not sure. I remember getting a cat on the girl version but I'm not far enough on the boy version for that yet.
Well, anyways. Oh, and theres a lizard that crawls around up by the waterfall, (it's on the side by the pond...thing...near the harvest sprites tree). Again, you can't do anything with this.