Always scale up your bee production. You can either mess around with soul frames etc. or just smash out enough bee houses and apiaries that you are always busy.
There are a couple of distinct steps in bee breeding that you should acknowledge.
Getting a mutation:
Do this in apiaries with pristine princesses. Have an Apiarists chest handy as dropping an unscanned bee in it will tell you if its a pure or hybrid bee, a task that otherwise costs drops of honey.
Start with a "pure" princess from species A, and a stack of (pure) drones from species B. You will, of course, quickly end up with princess and drones that are AA, AB, BA, or BB. Just keep feeding them back into the process: a princess that has become hybrid can be re-bred with a drone that is also a hybrid. (AB+AB yields AA, AB, BA or BB children, 50% of the children of hybrid parents will be hybrid and qualify for mutation).
Purifying a mutation:
If your princess and/or drones actually pick up the mutation to species allele `C`, then take them out of the mutation process. Its usually best to move the princess carrying a C allele and any drones carrying the C species allele to a bee house as further mutations cannot occur that - while interesting - might prevent you getting a pre-bred C line of bees. Here you need to examine your princess and drones - any drones that are xx (where x is A or B, but not C) are eliminated, but any xC/Cx bees can be fed back in assuming you can't find any CC drones.
Purifying / Generating a stable geneline
Once you have your first CC princess and CC drone, they will generate CC children but the other alleles will be all mixed up. But now you can move back to an apiary that you have got some pipes on. (Can't put pipes onto a bee house unfortunately, so in the very rare case that CC can cause a mutation you'd need to stick to a bee house and do this manually). You need to have a pipe/conduit extracting, another inserting, and then a low priority into a chest. Combs and drones will come out - if the input drone slot is empty, or has a drone they can stack with they will stack, otherwise the drones and combs will go into the chest. This will, after a few generations, create a princess and stack of completely inbred drones.
This device is useful in that you can use a stack of drones to quickly reprogram any princess to also be pure.
Only once you have a stable geneline archived do you take out a stack of inbred/purebred drones, and loop back to the "generating a new mutation" process.
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This process gives you the quickest route to a stable new mutation, but the resulting other traits going to be a mix of traits associated with species A, B and C. If you have other traits that are important to you, you need to work on producing a line of bees where you mix the different drone stacks you have stabilised in, and try and capture the best range, shortest/longest life, smallest/biggest fertility, best temperature/humidity tolerances etc with the goal of producing a line of drones with a species that doesn't matter so much as the other traits. Crossbreeding these drones with a princess will give you (obviously) a hybrid princess, but if you go through the geneline stabilisation process the resulting new line would have a good chance of carrying more of the better traits, especially if you carefully select the drones by scanning them and using up honey.
Actually, in a good number of cases, the allele you want from a mutation is not the species mutation, but something important like a good temperature or humidity tolerance - which are absolutely vital when breeding bees that don't tolerate "plains" type environments come into play. The species allele always determines the core "preferred" environment, but you can mix in up/down/both tolerance alleles so you can use the bees in any environment you set up your base in without resorting to (for example) Alvearies with environment modifiers.
