I'll try to explain what I did, it took me about 4 hours to get to imperial/industrious.
edit: THERE ARE TWO MAIN CONCEPTS YOU WILL WANT TO UNDERSTAND:
Mutating: Breeding for mutation is the pairing of two bees which will mutate when bred together. They have a chance at producing a "mutated" bee, which might be the next tier of bee. Bees that mutated when bred together can be found on the bee breeds wiki.
Bee conversion: By breeding a non-mutating pair of bees together, you have a chance at converting either of the bee into the other. This means you risk converting both bees into the type you didn't want, which is why I suggest stabilizing a pair before doing this. Stabilizing will be explained later in this post.
GETTING INDUSTRIOUS/IMPERIAL FOR ALVEARY PRODUCTION:
By breeding meadows/forest together you have a chance at getting common. What you want to do once you get a common PRINCESSES and a common DRONE, is to breed them together until they become a stable pair. This means that the drones will stack together after every life cycle. Once you have the ability to reproduce common drones, begin breeding your meadows/forest PRINCESSES with common DRONES in attempts to get the cultivated bee. Again, try to stabilize that species.
After that, by breeding a common with a cultivated bee you have a chance to produce the starter bee for either the industrious branch or the imperial branch. These are known as the noble and the diligent. Once you get a common/princess pair for either, begin to stabilize those.
By breeding the noble/diligent with the cultivated bee (similar to breeding common with meadows/forest) you can get the next tier of bee in either branch, unweary and imperial. By breeding unweary with diligent or imperial and noble, you may receive an industrious or imperial bee, respectively.
Tips for starting:
1. Stabilizing bee pairs will ensure you don't lose that species. Sometimes the mutation will go the wrong way (i.e. If you got a common drone and tried to convert a meadows princess to common by breeding them together, you might get a meadows drone & princess back, losing the common bee).
2. The more princesses you can get and the more apiaries you have running at the same time will greatly increase the chances you will get the mutation you want. When I started breeding I had 15 apiaries running constantly. This kept me busy the entire time, but I was able to breed multiple pairs of the precursors to the mutations I wanted.
3. Use frames early to get honey. Honey is essential in breeding, as it is required in the beealyzer. For breeding you will want to identify the bees that have the "shortest life" trait and continue to use those. You may want to find the "fastest worker" trait and continue to use those as well. By filtering out the "longer life" and "slowest worker" traits, you can begin to selectively breed your bees for success.
4. If you are in a snow biome, apparently snow bees have the 4x offspring modifier. This means they will produce 4 bees at the end of a life cycle. This is important because it'll increase the chances that a trait you wanted bred into the bee, will persist.
5. Most of learning how to play with bees is by experience. I didn't understand what most of this stuff meant until I mucked around in the mud for a bit. After a while you should get the hang of it and understand bees in your own language and develop your own strategies. Mine worked for me, simply because I'm a persistent bastard.
edit: You can replace the breeding of the common drone with any mundane bee (forest, snow, rocky, jungle, etc.)
