Getting traits onto an offspring is best done through breeding individual lone traits first. e.g. go for “swift” alone over “swift” “cold blooded” “hooligan” “serious” as it will be easier to later donate swift to a pal. Note that if you have two desirable traits without any fluff that’s even better.
If you don’t have it alone then breed that 4 traited swift pal with a blank traited pal until you get swift alone. Once you have a set of traits you can start combining them until you eventually breed those traits onto the offspring with the most positive traits without any fluff traits.
I’ve tried breeding two ways, one was generating novel traits by breeding two blank pals together as the game likes to fill in blank offspring with new traits, then breeding those new traits as outlined above. This can … take time. I got nearly 400 Pyrin Noct before getting all the traits I wanted. Then condense the traits you want to get the 4 “perfect” ones while also condensing the extra hundred trash pals into the perfect one for instant 4 stars.
The other way I bred was via ignoring what pal I had but focusing on the traits involved. What I first did was find 4 traits I wanted on a potential pal, this was for a battle mount so I chose Ferocious, swift, runner and musclehead. I did not care what pals I bred but rather only what traits the offspring had. I did, however, write the results for each pairing.
Over time I got a set of traits for each type of pal I wanted, a battle mount, a pure attacker, a tank, etc. From there I bred them to random blank pals until I created a new perfect pal for each grouping on the offspring. Eventually I had enough perfect traited pals and a listing that showed how I could turn, say a nitewing, into a blazehowl and I was able to breed up a perfect blazehowl battle mount in 1-2 generations.
Try consolidating traits whenever possible on a pal with a base egg (not large, huge or massive) type to reduce incubation time.
It seems work speed impacts breeding time, so it’s not a bad idea to start with your worker traits as this will also give you a more efficient breeding base when you only need one chicken, one cow, one bee, etc.
Make a dedicated base for breeding, I can get about 3 breeding pairs confortably while maintaining an on-site cake bakery. I suggest a base devoid of trees or rocks of any kind to prevent them mining/logging over working farms. This is especially important early on as you may not have the best choices for workers.
Don’t be afraid to combine low tier pals, it seems they just went for random results and not a curated list of results. For example a swee and sweepa made me a digtoise, it gave my base an early digtoise for mining. Or an early Bushi from nitewing and celaray.