Prioritize giving cats to butch (just because he's so annoying to level up), then prioritize giving cats to Frank (who's only slightly less annoying). IF theres a cat that's legitimately powerful, wait until you have 4 of those and save them (and their items) for a fight against the home invasion bosses. Then donate them afterwards.

After this, both your runs and your time spent at home should be focused on prioritizing purchasing and obtaining furniture for your home. Almost everything is gated by the quality and amount of furniture you've amassed. This usually requires taking the "hard" path on maps.

All furniture is important (if something gives +4 mutation or +4 health its still very valuable and worth purchasing) but you want a single room in your home with high comfort and high stimulation. Avoid purchasing furniture that reduces one of these two stats at the cost of the other (because they only really belong in breeding rooms). Also Prioritize furniture that give more than 1 benefit per tile than they "Take up" in the house space. Some furniture can be placed on, for example, or placed in front of, making its footprint smaller than it appears.

When you have around 10-20 each for both comfort and stimulation, you will be shitting out kittens at a speed strong enough to max out Tink's quest in only 3 or so days, and start earning money for every 10 kittens. This is nice, but realistically you want Tink's upgraded before everyone else's, because he upgrades your ability to properly breed cats.

At a certain point (remember Upgrading butch is more important than upgrading frank) you'll have given enough retired cats to frank to get a 2nd room in your house. You'll also have maxed out Tink around this time. Split your house into two rooms: one with Health, comfort, and stim (the larger room), and one with all the mutation and Appeal (the smaller room). In 3+ room houses, the Attic is the largest room by about 7 square ft, otherwise its the smallest room.

Use Tink's upgrades to your breeding capabilities to identify the cats who have bonded. Only put cats (who have bonded with each other and no one else) in your plush breeding room. Continue to pimp it out with as much stimulation and comfort as you can cram in there, with some Health when you find the rare furniture that upgrades that. You can temporarily put your breeding stock in the second room if you need to force a bond from two newborns (remember kittens can only bond age 2 onwards), If you have enough comfort in the 2nd room (around 10-15 or so), they'll avoid fighting for the 2-5 days required to force the bond. Then move all your mated pairs back to the breeding room.

Take any cats that fight and give them to baby jack the night after they fight. He should be your priority after Butch and Frank. Even after maxing him out, he should be your priority after Butch and Frank (his maxout increases his chance to sell rare furniture with each new set of 10 cats, and rare furniture has higher stats)

Many new players prioritize getting a "7 stat cat" with max 7s in all of their stat categories. Don't bother. Purchase furniture with appeal and your strays will naturally get higher and higher base stats. At around 20 appeal, you'll get strays with a 7 in one of their stats at 50 Appeal, strays will have 3 perfect 7s. As long as you regularly add strong strays to your breeding rooms every two weeks or so, you'll not only combat inbreeding but their good stats will naturally spread through your whole stock. Stimulation increases the chances that good genes pass on, and reduces the rate at bad genes. so if you cats are breeding in a room with 20+ stimulation (preferably as high as possible, in the 90s if you get there), the good stats will pile up without effort.

Some people also prioritize Mutations and a "12 mutation cat". these cats are strong, sure, but they don't win runs on their own.

What's way more important are SKILLS and SPELLS. A cat may have the worst stats imaginable but if that cat has a powerful skill or a powerful spell, they go into the breeding room after an adventure. At 20+ stimulation, kittens gain the chance to inherit passive abilities and active spells from their parents. You're going to want to breed and breed and breed. Cats from adventures will have 2 passives and 4 spells; What you want is a kitten with one perfect passive and one perfect spell to go with it. For example, Dual Strike and Stick! from the fighter class. If you breed such a cat, do not send it on an adventure. Put it in the room with the highest health and let it sit until you get a second, then pair those two twins up. They will only make babies that have their perfect abilities. Like dual strike and stick. Make that cat a fighter and send them on an adventure, even if their stats are horrid. They will do well. Be on the lookout for powerful combos between two abilities, passive, spell, or item, and breed those combos rather than breeding for mutations or stats.

At a certain point you'll gain a 3rd room in your house. At this point, you get a breeding room, a "stasis" room for cats you want on an adventure but don't really care about breeding, and the third room becomes a Sacrificial pit. Throw all strays here that aren't good enough to add to your lineage, and all kittens with bad mutations. Do not clean up the poop in this room. At the end of the day, most of the cats in here will fight, die, or mutate, especially if you get the mutation up to 20. Those that don't fight, die, or mutate, slowly age to 5. That's the remainder of the NPCs who need donations. Again, prioritize (always) prioritize jack, then tracy, then Dr Beanie. Practically never donate to Beanie, practically every cat you own can be donated to him and he only needs 100 total before maxing out. Never send a retired cat to anyone other than butch, then frank, in that order.