If you have a dog in your party and check its details, you now have the option to pet it.
If you want to pet your own Dog, the fastest way is through a couple simple fusions. Dogs don't show up as enemies (and so aren't recruitable) until late in the game due to them having the valuable Encourage ability that raises all of an ally's stats.
To start making your Dog, you'll need to first fuse any plant monster with a Fairy found in the second level. This will get you a Wishing Seed, which also normally appears later in the game. Fusing any beast monster with that Wishing Seed will result in a Dog who is 100% guaranteed to be a good boy.