Despite all the awesome witch and zombie and werewolf action, I find myself drawn over and over to the beekeeping trade. I would not call it a hobby, because there is no skill involved.
To get started with beekeeping, simply buy the beehive (or use the beehives that are found behind the alchemy store) and get to work. The beehive can be found either by looking through Collections -> Supernatural, or Sort by Room -> Outdoors -> Outdoor Activities.
The hive comes filled with bees and comb honey. You have four basic actions your Sim can perform: Smoke Hive, Clean Hive, Feed Bees, and Harvest Honey. Before your Sim works on the hive, they will change into a beekeeper's hat and veil.
You will need to work on your hive about once per day. A wise beekeeper smokes the hive before performing any other action. As in real life, smoking the hive makes the bees sleepy so that you are less likely to be stung.
Note: if your bees are happy, you may not need to smoke the hive first. The first few times I sent my Sim to work on the hive, I did not smoke it first, and she did not get stung.
To successfully keep a hive, you should feed and clean the hive once a day. If the hive is neglected for too long, the bees become angry and will be likely to sting or swarm attack.
There are two negative moodlets associated with bee attacks: Bee Sting (-30, 24 hours) and Bee Attack (-200, 15 minutes). Being attacked by a swarm of angry bees is no fun! (For the Sim. It can be fun for the player, depending on how sadistic you are.)
You will be able to harvest honey about once every 24 hours. When you harvest, your Sim will end up with both honey and beeswax in their personal inventory. I found that harvesting from the hives behind the alchemy shop gave my Sim much more honey than her hives at home. But that might have just been a fluke.
Both honey and beeswax are used in certain alchemy recipes. Your Sims can also drink honey straight, or use it in cooking. I have also heard that beeswax can be used for candle making, but I have not explored this yet, so treat that as strictly rumor status at this point.