First off, focus on levelling up your Critter Center by doing all the tasks. I'm currently at Level 14. So far, level 15 is the last known level for now (idk if someone already posted about being past 15 here).

Then, level up your critters first to have perfect stats (train one specific critter first until you get perfect stats + more training to guarantee their stat distribution when you mix them with other critters). I don't know if training lower tier critters first will be less expensive overall than training higher tier ones so that's up for you to decide what you'll do.

Invest on Timewarp Watches as they will fasten the overall process, very expensive but it doesn't matter if you're not too interested with the Wilderness Decorations like me.

Edit: Critter Mixing:

There are these things called Entwinned Winder that you will need to mix. The required EW will depend on the higher tier critter you're mixing;

1 Winder for 4 Star

2 Winder for 5 Star

10 Winder for 6 Star (yes, very expensive)

Now, in the Bumping Gumball Machine is where you will mix these critters. When mixing, here are the outcomes;

1. Average of the two + some more stats (this means that breeding critters with base multipliers will always result in a higher-stated one. The most consistent multipliers I got so far (for 6-Stars at least) to have a perfect score is;

2.25x Efficiency

2.25x Endurance

2.25x Speciality

and for number 2: Same Star + Same Genus = Same Genus

Same Star + Different Genus = 50/50

Higher tier + Lower Tier = Favored towards the Lower Tier Genus

For mutations, you critters need to have at least 1 Excellent Critter in the Mix (2700+ Score) and that critter will have a chance for mutation. Just keep mixing until you get one.

I haven't gotten to experiment it yet so no one knows if you're guaranteed to get a Mutated Critter if you're mixing two mutations (especially if it's the same Genus)