Breeding in ARK is quite a complicated mechanic when your goal is to get the best creatures. I will try to keep is as simple as possible.
Selecting the parents
The first step in breeding creatures is to carefully choose the parents. You need to have a look at their base stats, before any level up and before any imprint bonus because the offspring only inherits the base stats of the parents. There are different ways to check those:
Your creature has not levelled up yet, and has been tamed/does not have any imprint bonus. The stats you can see are the stats that can pass on to the baby
Your creature has already been levelled up and/or has an imprint bonus. In this case, I would recommend using the ARK Smart Breeding tool. This allows you to extract the stats of your creatures, be it a fresh tame, a fully levelled dino, or an imprinted creature
Either way, I would highly suggest to keep track of the stats of your different creatures. I personally use a spreadsheet where I write down the stats of the creatures that I am interested in breeding. This is quite basic though, as it only highlights the best stat I managed to get so far.
ARK Smart Breeding allows you to create a library of your different dinos with their respective stats, and can also guide you with a breeding plan. Each time you extract the stats of a creature, this tool tells you how many points it has received for each stat, giving you an idea if it is high or low.
For each stat, the baby has 55% chance to inherit the best value either from its mother or its father. Example: Mother has 1000 HP, father has 1500 HP. There is a 55% chance that the baby will inherit the best stat, 1500 HP. This is the stat selection process. Sometimes you will notice a slight increase in a stat, this is only because the game considers that a stat is passed on a baby based on a 100% taming effectiveness bonus which you may not have achieved with your creatures, thus sometimes slightly increasing stats.
I do not have any confirmed numbers for it, but I am assuming that there is 50% chance for each colour of every active colour region to be picked during the colour selection process.
As you can see, breeding is all about luck and patience. Unless you are very lucky, this will probably take you a few generations before you manage to gather all the best stats in one individual, or even better, a male and a female which would guarantee these stats for their offspring.
There is very little information about how mutations work in ARK, and to me the whole mechanic seems a bit messy at the moment. A bit of information first:
Mutations seem to have 2.5% chance to occur
Mutations are only positive at the moment – you will not get penalties from inbreeding
Inbreeding does not seem to increase the chance for mutations to occur
One mutation seems to mutate one stat and one colour at the same time
A mutated stat gives +2 levels
Mutation score = patrilineal mutations + matrilineal mutations
/!\ TBC: A creature with a mutation score of 20 or above will not be able to mutate anymore
As the chance of getting a mutation is pretty low, your best bet is to breed a male that has the best stat with as many females as you possibly can, until this stat mutates, preferably on a male. When a mutation occurs, the increased stat is considered as a base stat. This means that you can go on with this new male, until the same stat further mutates, and so on. Theoretically, once a creature reaches a global mutation score of 20 or above, it is not able to mutate anymore. If you ever managed to breed two creatures with a mutation score greater than 20 and the baby mutated, please let me know. Though if you breed it with a creature that has a mutation score below 20, then you could in theory mutate a stat over and over again.
There are some cases where you are not able to see a new colour and/or an increased stat on your creature despite it having an additional point to its mutation score. This could be because the new colour has been applied to a colour region that your creature does not have, as a result the colour mutation is lost and you do not see it. As for the stat, this could be because the mutated stat has not been chosen during the stat selection process, or because movement speed has been mutated, which does not impact the actual speed, but still grants a mutation point.
Two of these issues will be addressed at some point by the developers: the fact that a mutation can occur on a colour region unknown to the creature, and the fact that movement speed can mutate. I will edit this guide when they put those changes in place.