The importance of atmosphere
If you want to roleplay, of course an RP server is the place to be. If you want to try out RP for the first time, you may transfer your existing character to an RP realm, or else start out afresh with a new character there. Both are good choices, depending on how much you are willing to leave behind, and how much time you are willing to invest in your new RP life.
Of course you will also be wondering which server will be best for finding roleplayers. Indeed some RP realms are filled mostly with normal players, who don't actually want to roleplay. Some of these players didn't know what a RP server is when they made their first character, and some just play there because that's where their friends are. Still others used to roleplay a lot but have since lost interest for whatever reasons. Either way, you may be sorely disappointed if you spend the time and energy establishing yourself on a new RP server only to discover it doesn't have so many roleplayers on it.
Finding the "best RP server" is hardly a simple task, however. Some would argue that any realm would do as long as it had a good circle of friends for you to RP with. While this is true, of course, some RP servers definitely have a more lively RP atmosphere than others, and this makes a big difference. The general atmosphere of a realm can either make your circle of friends feel like members of a grand RP community, or outcasts living in exile, the last bastion of RP civilization.
Birds of a feather
This "New Person's Guide to RP/RP-PVP Realms" on the official Roleplaying forums is a good place to start. Plagued by countless questions about the "best RP server," its author decided to create one thread in which players from all around could "rate" their own server according to various criteria, so that people could not only get a sense of "how good" it is for each faction, but also get a feeling for each server's particular RP feel, culture, customs, and so on. My reading through this thread led me to choose Moon Guard as the home for a couple of my new Horde characters, and I remain very satisfied with this choice - It's very pleasant to walk around and see a majority of people actually roleplaying wherever you go, particularly in Silvermoon City.
A while back WoW Insider also featured a couple posts in which readers left comments about which roleplaying realm they felt was the best, recently enough that the situation probably hasn't changed much in the mean time.
You may also, of course, ask any roleplaying friends you might have as to which server they like the best and why. Be sure to clarify which faction your friend has experience in on that particular server, as sometimes the Alliance and the Horde community vary widely in the quality of their RP atmosphere.
Someone to talk to
Whenever you narrow your list down to a server or two that seem good to you, check out their realm forums to see what the people are saying there. Most RP realms have a stickied post at the top of the forum page which tries to list all the prominent guilds and give a short description about each one. In the best cases the description is accurate and interesting, and there is even a link to a proper web page; but in the worst cases some entries on the list may be quite out of date, and some guilds may not even exist anymore. Remember, the intention here is not so much to find the best guild for life, but to give you some names of people you can look up to get started meeting people and interacting with them in character - though I once found a wonderful roleplaying guild for my first serious Horde character on such a list, and have been very happy with it ever since.
I must caution you against following anything other than these stickied resources on server forums too seriously. Lots of people use realm forums simply as a way to insult each other or advertise something. Although a great portion of good roleplayers don't even bother with server forums at all, you may get lucky and find a few helpful people who are willing to point you in the right direction if you're having trouble finding a guild that looks interesting. You might put up a post saying, "Hi, I'm a new roleplayer on this server - would anyone like to come meet my character?" and see what kind of response you get. Ignore the trolls.
Of course it's always possible to just start up a character and head out into the world without knowing what you will find. This way is, unfortunately, most likely to lead to disappointment, since you may play for a long time in the company of people just mindlessly leveling alts rather than coming into contact with real roleplayers.
Meeting roleplayers - hopefully being surrounded by them - is the first priority when picking a server. But in the end it is not enough; you must also find a home for your character where you can connect with real roleplayers on a long-term basis, in many cases making friends you will enjoy for a long time. Next week we'll look at how to find and keep the right RP guild for you.