To speed up the breeding process you can use chocolate frames (check JEI on how to craft these). These frames speed up the whole process because it will kill off the queen far quicker. Instead of waiting a few Minecraft days, you can kill off a few queens a day. This enables you to breed faster, but does decrease the amount of honeycombs you get.
Oblivion frames can also be used to instantly kill the queen and give you offspring. These frames are expensive, so use sparingly.
You will have lots of bees after the breeding process that are not useful. Either they have the wrong traits or did not become a purebred (Forest - Meadows instead of a Common - Common). Don't throw these away as they will become useful later on. These might be hard to store, but keep putting them in chests. You can throw these away if you have no space. Only keep them if they carry the bee species gene you are trying to get.
Once you have a good supply of Common and Cultivated Drones, you should start breeding a few more Common and Cultivated Princesses. This will make life easier. Generally, it's good to have 2 or 3 of each species of Princess as there is always a chance to lose a Princess or a mutation gone wrong.
The easiest way to get these extra Princesses is Brute Force Breeding. Once you have a healthy amount of Common and Cultivated Drones you can drop these into a Apiary with most Princesses (Rocky are best as many of them are Pristine Stock) and keep feeding whatever Princess comes out into the Apiary again until the offspring stack with the original Drones you put into the Apiary.