Note: Gendustry machines are well-documented - go check out the website for more detailed instructions on how to use them!
The first machine you’ll need is the Mutagen Producer. This will convert certain materials into liquid Mutagen, which is required to power the Mutatron. I usually use Redstone Blocks for this, but it does accept glowstone and other materials. It will also require FE to operate.
The Mutatron is the second most important machine. It will consume Mutagen to force a mutation and is a huge time-saver. It also bypasses all mutation requirements. If you can afford to do so, I strongly recommend immediately upgrading to the Advanced Mutatron, which will give you the ability to choose your resulting mutation, as some species have multiple results. Note that the mutation process has the chance to kill ignoble princesses. I haven’t run enough tests to guess at the rate of deaths, but it doesn’t seem too high.
Another important set of Gendustry machines is the Sampler and Imprinter. Basically, you can sacrifice bees in order to get a copy of a specific trait, which you can then use in an imprinter/template to alter bees. So, if you want your bees to act faster, you’d want to obtain bees with the fastest trait (you can look these up in JEI). If you want ones with a larger territory / range, produce some of those. Each bee sacrificed will produced one gene randomly out of whatever active and inactive ones it possesses. You can sacrifice princesses, but it’s a waste. Gene samples can be recycled by smelting them.
When you have gene(s) you like, you can add them to a template simply by crafting them (one-by-one or multiple) together with a template. The gene is consumed and permanently added to the template. Blank characteristics in a template will not change those characteristics for the target bee. So for example, if you wanted to make your Rocky Bees have 4 offspring instead of the default 1, but keep everything else the same, you could sacrifice Majestic drones until you get the offspring: 4 trait, then add it to a template. If you were to use this template, every characteristic would remain the same except for the offspring, which would take on the new 4-offspring trait.
The Imprinter is the machine that uses the templates. Each imprinting consumes a labware and has a chance of killing an ignoble princess. You will need to do this twice for each species you wish to modify (to create a breeding pair).
If you want to copy samples or templates, you can use the Genetic Transposer.
This set of machines works together to create new (ignoble) princesses. The DNA Extractor requires sacrificing organic materials (drones, saplings, etc) to produce Liquid DNA; the Protein Liquifier creates liquid Protein from raw meats. Both of these are combined in a Replicator, along with a full genetic template, to produce a new ignoble princess.
I would start making use of the Mutagen Producer and Mutatron as soon as possible. There are a couple of pairings that lead to multiple mutations, and being able to force a specific mutation takes a lot of RNG frustration out of the bee breeding process. This will let you proceed through bees pretty quickly.
After you've gotten a couple different species to play with, you'll want to start looking at the Sampler and Imprinter to make more of specific kinds of bees, or make them all nocturnal / cave-dwelling / rain-tolerant, so you can have your bees running 24/7 anywhere you'd like. I haven't yet found the need to make more princesses, since there isn't a whole lot of competition for hives, but it's definitely an option to consider if they're hard to come by.
So if you’ve been going the Gendustry route, you’ve probably already made a few of these already. There are 4 slots in the middle of each Industrial Apiary which can be used for upgrades. Some allow multiple copies of a particular upgrade; each upgrade (including duplicates) will increase the RF cost. I will cover the ones I use the most here, but a detailed list can be seen here or in JEI:
If you’re doing Industrial Apiaries (Gendustry), it isn’t really necessary to know much about these. Frames are basically the equivalent of upgrades, with some reducing lifespan, increasing production, etc. There are no frames that completely change the climate or daytime / sky / rain requirements; for this, you are expected to gradually breed more tolerant bees by crossing them with species that have the desired traits, or use an alveary. Binnie’s Genetics also provides the ability to modify the heat/humidity tolerances. Alvearies are a multiblock apiary that has a natural +100% production boost, and the ability to modify the climate with special alveary blocks.
It’s been a very long time since I’ve delved into Binnie’s mods, aside from some of the bees, so I’m not comfortable talking about it in a guide. The machines have been revamped and the principles look pretty cool, but I just haven’t taken the time to explore them yet. From looking at the wiki and from what I remember, there are similar options to tailor specific bees to your liking by getting specific genes and acclimatizing their temperature tolerances so you can breed bees in a particular biome. These can also be covered using Gendustry machines (Sampler/Imprinter, throwing a biome upgrade or temperature upgrades into the Industrial Apiary), so if you are going the Gendustry route, you probably won’t find yourself using them.