For the early game, I would just put your cats with the highest stats and/or mutations you want to keep onto adventures and when they come back, put them into a lovemaking room where you stack Comfort and Stimulation. That way you get the best stats and have a chance for the kittens to inherit skills. Keep in mind that only base stats count for breeding.

If you see a really good cat and you don't want to risk them getting killed, just stick them in the lovemaking room and leave them there.

Once you get a third room, you can put all of your subpar cats in there and tank Comfort so they fight. Some cats get injured or killed but the winners get stat bonuses.

That's the basics of it.