I have rode through the world of hops like a Horseman of the Apocalypse, leaving a path of fire and death in my wake.
Hops are widely believed to be a hardy breed and, once established, can be very difficult to remove. I challenge this assumption by offering up the wrecked bodies of numerous hop plants I have killed.
Before you rush out and buy a bunch of hop rhizomes, the big question you need to answer is, what, really, do you want to accomplish with your fresh hops? The advantage to pulling hops right off the bine and tossing them into your wort is you will get a fantastic fresh taste which most commercial beers can't even match. It will also give your brew a sense of terroir which beer does not normally possess.
However, unless you have the equipment, homegrown hops are hard to dry and store. Also, without a lab analysis you are going to have no idea what the alpha acid levels are and so it will be impossible to accurately formulate recipes and calculate IBUs. If you are a better brewer than me you may be able to do all of this by taste but I can't so I like to make one or two fresh hop beer a year and the rest of the time use pellet hops. I can accomplish this with one or two plants.
Many homebrewers dream of having a few extra plants and selling the additional hops to other homebrewers or hop processors. According to Niko from Nikobrew.com, there is really not much of a market for small production hop farms. If you are interested in selling your hops I would contact your local brewery or homebrew club to see if you can get something going but don't expect to quit your day job to adopt the swanky pimp-like lifestyle of a hop farmer.
Anymore, you can buy hop rhizomes directly from places like Freshops.com or most large online homebrew supply stores. Morebeer.com has them. The selection is somewhat limited to open source plants. Varieties like Citra and Simcoe are proprietarily owned so you wont be able to grow your own.
The rhizomes are harvested in February so they will be available for planting in March and April. If you want to expand your own hopyard by splitting apart an existing rootball try to leave the majority of the roots intact. If you just chop it up you will kill the plant.
Once you pick out the hops you are interested in growing, buy two rhizomes. I experience around a 50 percent kill rate. You are supposed to plant the rhizome in sandy soil with the buds pointed up. I never really figured out what this meant because the rhizome looks like a stick and most of the time it doesn't have any noticeable buds. Plant both of the rhizomes in one hole, horizontally. If you don't have a ton of room they can be planted in a five gallon bucket or large pot but this will not give the roots a ton of space and you will not get great yields.
Ideally, hops should be planted in sandy soil to reduce standing water, which will cause downy mold and kill the plant. Hops like lots of sunlight, nutrients, frequent watering and walks on the beach. Pretty much like any other plant.
I have had some issues with aphids and grasshoppers. I normally try to spray everything with soapy water and then give up, when it doesn't work, and switch to the deadliest pesticide modern science can provide.
What not to do
When one has been brewing and drinking all day, it may seem like a good idea to dump your trub on your hop plant. This will kill it. I don't know if it is the change in soil pH, or if hops refuse to cannibalize, but your plant will die a horrible death.
I've done this twice. The first time I dumped hot wort on my hop plant and it was totally dead the next day. The second time I let the wort cool and then poured in on the hops. Then the all the leaves fell off and most of the bines dried out. It did kind of rally but was pretty sick for the rest of the year and never yielded any cones.
On found hops
Nearly every brewer has, at some point or another, come across some random hop plant growing in a friends back yard or along the fence of an abandoned farm. I've done it and gleefully harvested a bushel of hops out of an irrigation ditch, dried them and brewed a beer. Guess what? It tasted catty like I had filtered the beer through a meth whore's merkin. If someone told me they had found a sack of mystery grain in their basement I would politely decline brewing with it. Actually, I would probably taunt them as dipshits because I'm kind of a jerk but the point remains the same. Using fresh hops is difficult enough without adding in the additional complication of having no idea what kind you are dealing with.
Fresh hop beers are great and can be another brew in your seasonal brewing schedule. But don't expect your backyard hop plant to supply all of your hop needs.