For 3DS systems, Nintendogs+Cats. Or if you can find the original Nintendogs for DS that too. Basically you raise virtual doggies (or cats). You can train them, take them for walks and enter contests, doggie stuff. In terms of collecting, it would be hard. There are variants of items like leashes and some other stuff, but you have to get those gifted to you by Streetpass I think with other players who have a game different from yours. For example I have Chihuahua and friends. In order to get a different leash colour I need some one from Dalmation, Lab or another breed game to gift me that item. But other than that, it's a cute game, not meant for long play periods, can be boring if you play for too long in one sitting.
For Wii or Switch, Go Vacation. For the Switch it's basically the same game as the original Wii version. I super love this game. You play a bunch of mini games like horse back riding, racing sleighs, ATVs, shooting clay disks, kayaking, dog sleigh, scavenger hunts around the island. If you play enough games you unlock stuff for your villa, your house basically. If competing against CPUs arent your thing, you can complete little goals like 'sprint on horseback for 1000 meters', complete stunts and do flips on your ATV, go find all the photo spots, find all the off map locations. In the Switch version they added the 'find all the resort animals'. Each resort has animals that live there like hawks, reindeer, polar bears, toucans, beavers, ect... It's really good if you arent into competition, plus it still gives you stuff to do.