Aside from the starter pets that will be available to you at the beginning of the game, Starbound also offers a wide range of animals that you can capture and claim as your own pets. Home planet pets are more suited for companionship and following you around, while the space monsters are more for combat purposes, so it’s up to you which types of pets you want to go for.
Before you can capture pets, first you need a capture pod. Capture pods can be crafted at the Creature Capture Station. Do note that you can only capture small animals with a capture pod, and they have to be below 50% HP. This means that you can’t capture adult monsters or other NPCs.
Once you’ve successfully captured and tamed them, the pets will follow you around wherever you go. It’s likely that they’ll get stuck in obstacles or even get lost, but once you’ve ventured far enough away from them, the pets will respawn at your location. So you don’t have to worry about them being lost in space forever. You can also leave your pets in your spaceship, and they can be taken to any Starbound server as well, as long as you’re using the same character.