To unlock slugpups, you need to finish Gourmands Food Quest. Basically, eat almost everything in the game. Theres a food tracker above your hunger to help you. These are all the foods you need, from the wiki.
After completing the food tracker, you have to get the Outer Expanse ending.

Before you raise a slugpup, you have to get one. To find one, they spawn in shelters in regions. Each region has a different chance for a shelter to contain a slugpup. For example, you shouldnt look for one in The Exterior, because it has a 5% chance. Easier regions to look for them are Pipeyard and Shaded Citadel. Pipeyard has a 50% chance for a slugpup, and Shaded Citadel has a 40% chance. (dont get them confused like me lo) The best region to look for a slugpup would be Industrial Complex, as it has a 25% chance and a high amount of shelters. The region is also pretty small, which makes it easier. !!!NOTE!!! Slugpups are only available (without mods) as Survivor, Gourmand, and Hunter.

When you first meet your slugpup, in or out of a shelter, they will be afraid of you. You can tame them by holding their hand (grab them), or giving them food, if they dont let you grab them. (then you should be able to grab them) After holding them, you can hold the grab button to put them on your head, where they wont weigh you down. You dont have to do this, but i recommend it. Then, take them to a shelter and sleep.

After sheltering with your slugpup, you should notice a smaller food bar above yours. This is your slugpups food bar, and they need to be fed every cycle. They cannot eat corpses, even if your slugcat can. Slugpups like certain food, and dont like some others. Experiment with foods to see what your slugpup likes. If you feed them stuff they dont like, they wont like you! Slugpups cannot hold extra items when held. Fun fact! Idle slugpups might choose to take a nap. Less energetic slugpups will do this more often, while energetic slugpups will tug on you while idle.
When in danger, their food bar will start blinking red. When dead, their food bar will shrink and turn grey.

Beware of child murderers, like Caramel. He killed my son, Pearl. To protect your slugpup, keep them away from scavengers, and DONT be like artificer, hold your slugpups hand or carry them when going through tolls.
If anything kills your slugpup. ANYTHING. Then you kill them, and die, or restart the cycle by quitting and going back into the save.

You can train slugpups to see certain enemies as threats, and some as prey. Slugpups will more commonly kill enemies that you kill often. They can also somehow kill things better than most players, even though they have the damage of the monk. They will also move around like you, so dont teach them to jump straight into enemies, that wont be good.

You can also tell your slugpup commands.
-If you jump while holding down, the slugpup is told to stay in their location.
-If you jump once without moving, the slugpup is told to play with the player. In this mode, the slugpup will stay closer to the player.
-Jumping while holding up tells the slugpup to stop playing with the player.

Slugpups have six different personality traits.
Sympathy affects aggression. Bravery affects nervousness, aggression, and dominance. Energy affects nervousness, aggression, and dominance. Energetic slugpups also want to hold pretty or shiny things (pearls, overseer eyes, and vulture masks) Energetic slugpups play more often, and less energetic slugpups nap more often. They also pull and tug on the slugcat when idle.
Nervousness is affected by bravery and energy, and nervous slugpups are anxious around other slugpups. Aggression is affected by energy, bravery, and sympathy. It affects dominance, and aggressive slugpups hold weapons more often. Dominance is affected by energy, bravery, and aggression, and they sometimes hold things like noodlefly eggs. Dominant slugpups stick with other slugpups, and pick up items more.

-You can feed slugpups drugs. They start shaking for a bit when you feed them drugs, and thats all.
-You can feed them to lizards. Why the hell would you do that though? What kind of monster would even do that?
-Slugpups cannot ascend, which means you have to leave them behind on hunter runs..
-The ending for Survivor changes depending on how many slugpups are brought. Proven by the monks OE ending, survivor brought 2 slugpups.
-Violet and red are the rarest slugpup colors.
-Slugpups can be either dark, or light. Light slugcats have dark eyes, and dark slugcats have light eyes.
-Giving slugpups items and gifts (such as pearls) will make the slugpups like the player more.
-Slugpups will play and fidget with items in their hands when idle.
-Slugpups can point.
-When a slugpup dies, their body will turn a shade of grey slowly, like Saints freezing body after death.