(Totally rewritten, 8 Jun 15 -vKT)
There are 2 types of rabbits - white and black. They are an excellent source of coin (white 5k, black 10k), experience (500), collection items (incl. rare Mining & Fisher sets) and the black also drops 50% fruit tree fertilizer. As with all animals, they start out as babies (called "young") for 2 cycles, then become adults. Currently, they cannot be contained with any fence so if you don't feed them they will eat all the grass/brush at whatever station you have them (see FOOD). A quest will give you 2 white rabbits to start with, but can then be bought in the market; black will become accessible through the Nameless Forest quest line.
They can also get seriously out of hand if they are not managed well, causing game lag and general ravagement of your station and food resources. There are 2 ways to manage them so that they don't get out of control (see STRATEGY). The 5th stage of their 25 total cycles is propagate/breeding, which puts a heart bubble over their heads. That stage will last for 8 hrs before they move on to their next cycle. Young sell for the same price as adult.
If you allow your rabbits to eat the grass/brush from a station, you will lose all the goodies you'd have gotten if you'd cleared it yourself. The plums, honey and apple energy items, worms (which will end up being very important later) and frogs (make a +5 energy item in Polar Side) are very useful. Admittedly, if you allow them to eat that stuff it does save you a lot of energy clearing it yourself, but even then, at some point the natural food will run out and you'll need to feed them yourself.
Rabbits will eat grass items and hay. You make hay in the barn from grass and wind, then into bales. There are several recipes for bales, but all except 1 use wind or water at some stage. However much you think you have, it will be gone faster than you'd think possible if you use it for feed.
The solution for this is one of two items - the Fountain for Coins or the Fountain for Emeralds.
The former is bought with game coin and includes 1 recipe for grass and 1 for jasmine bushes (which are classified as grass). They are both 5-count size. While both do use grass in their recipes, if you were to make it, place it and then chop it, it does produce more grass than it used to make it. The jasmine bush does not auto-place from storage, meaning you have to go back into storage and get another one to place each time - I found this unnecessarily time consuming.
The latter is, in my opinion, the more long-term solution. The Fountain for Emeralds is bought with emeralds and has 1 recipe using grass to make 10-count grass, 1 recipe using grass to make 15-count grass, and 1 recipe using cabbage to make 20-count grass. (It also has a 7-count bird food recipe, but I don't feel it's very useful). The last recipe uses cabbage, which is quick to grow and not heavy to transport, and water. Being 20-count also means you spend less time tending to the rabbits.
There's really only one way to handle rabbits in order to maximize profit without causing the game to run like a pregnant hippo and annoy visitors, but there are 2 paths you can take.
1. Starting with 2 white rabbits, when the heart bubbles come up, click the bubble and it allows them to propagate. They will have one baby, called "young" and it will be pink for the 2 cycles it remains young (young black rabbits are gray). Once it becomes adult, you'll no longer be able to distinguish it from its parents, so it's easier (and more food-efficient) to sell them when they're young. On your in-game menu bar, find the gold arrow and click it, then select the red X to sell, target the pink/gray rabbit(s) and click.
Obviously you'll keep the first few until you get the size herd you're comfortable with. You'll also need to pay attention to the cycles, so that when the adults are on their last stage before turning gold, you keep the most recent young ones to become your next herd.
2. There are two items that can make the process seriously easier to deal with - the Rabbitry and the Large Rabbitry. The former handles only white rabbits, and the latter will handle both. For both types, rabbits in the Rabbitries are effectively on hold - they will not eat, breed or expire. The Large Rabbitry will hold up to 98 blacks (I haven't tested with both colors yet). I haven't used the regular Rabbitry, but I would assume it's the same limit.
Step 1: Breed to the size herd you want, when you get your first batch of young ones, click on the Rabbitry and pick Add 25 Young. This grabs all the pink/gray ones and puts them in the Rabbitry.
Step 2: Go do other things until you feel like coming back and letting them propagate again.
Step 3: Once you have a new batch of young, click on the Rabbitry and pick Sell All, then click again and pick Add 25 Young.
That's it. By letting a new batch propagate before you sell, it insures that if you come back and find all your rabbits gold statues, you simply Release All on the Rabbitry and let those become adult. You'll never have to buy another rabbit, which will drastically improve your profits. For me, 25 or so at a time is plenty, doesn't cause me lag (although you still don't want to put rabbits near your entry tunnel where neighbors arrive) and doesn't cause me a lot of effort to feed. I place 10 or so 20-count grasses a day, if I've let them breed most of the times they were ready.