Whether new to roleplaying or experienced the first step in having a good experience is scoping out the roleplaying community on a new MMO.
Visit a game's official forums. You'll want to see if there are any special forums for roleplaying, player fiction or discussions about the setting. Check the stickies to see if there are any special listings for roleplayers or roleplaying guilds. Read through a few threads, lurk a bit, and get a feel for who is who. Many folks will include links to their own guilds in signatures and if you see something interesting go have a look.
If you're very lucky there will be a very strong roleplaying community on a particular server and, very often, these communities will run their own forums off-site in order to better police content and create a comfortable, productive, environment.
Keep in mind that many roleplayers can be outspoken or opinionated to the point of being obnoxious. Don't really let that put you out. Just keep browsing until you see posts you find interesting or enjoyable and make note of who is behind them. Many discussions will be OOC and will pertain to how gameplay relates to the setting or what approaches roleplayers can take to simulate some iconic situations.
Others will be very silly. Silliness, OOC, is a popular tool among roleplayers, even otherwise gruff ones, because it can defuse quite a bit of the occasional, above mentioned, obnoxiousness. Hey, we take things very seriously sometimes. That's good to a point, it means you care about what you're saying, but sometimes it descends to typical forum epeen waving. That's when it's time to break out the lolcats, baby!
Okay, so there isn't any clearly great RP server and the forums aren't all that helpful. How do you find the roleplayers once you're in the game itself?
Some games include a 'Roleplayer' flag. Make sure to turn yours on. Even if you're not roleplaying at the time it's at least a basic advertisement to other roleplayers. It can, however, attract anti-RP griefers. These are usually best handled by using the ignore command (/addignore, /ignore). If they persist, actually hamper you in some way, just take some screenshots and file a report with the GMs. As a rule don't respond to them. It's what they want. Also keep in mind that, in some PvP oriented games, there's a fine line, or no line, between griefing and acceptable gameplay. On the other hand, some games do have official RP Servers and rules for them. Know those rules but only file complaints when you really have to.
Now your tag is up, you're prepared to deal with the rare griefer, and looking to meet folks. The obvious bet is the local watering hole. Taverns, cantinas, bars and clubs are where you'll find RPers hanging out. In some games roleplayers primarily roleplay only in group or guild channels and you might see people standing around and not saying anything. In others, everyone roleplays in 'say' (or spatial) and you can even find yourself getting spammed out and unable to keep up.
The best advertisement for your own roleplaying is to actually be roleplaying even if it's by yourself. Describe how you enter the room. Don't pose massive text blocks though. People in crowded roleplay situations tend to see that as spam or attention-whoring. Make it short but vivid, make it say something about your character. One sentence or two shorter ones should be all you need.
Wait a while. Try and get a handle on what's going on, if anything, and look for other people with RP tags.
Some games also include text-biographies for characters. You can right-click on the character in question and see who they are. Now, this is usually only information for you, the player, not your character. There's no way your character knows that other character's life story or, even, their name for that matter. Keep in mind that what you know and what your character knows are two different things. Roleplayers call this IC (In Character) and OOC (Out of Character) knowledge.
Exceptions to the OOC Only rule for biographies can come in a few forms. Physical descriptions are IC knowledge as the player is trying to flesh out how a character looks, sounds or moves beyond what the graphics of the game can express. In high tech settings there's sometimes assumed to be a central database or internet of some kind and players might post what your character could find, ICly, if they looked them up. You'll be able to tell as these entries are usually laid out as official forms.
At this point you have a few options.
Send a /page or /tell to one of the RPers and introduce yourself OOCly. OOC communications are usually framed with a double paren like this: ((This is an OOC communication.)). Why double parentheses? I have no idea but that's the usual convention. You'll also see brackets and single parens. Many players, also, assume all /tells are out of character but it's best not to assume that. And to give you even further pause, there are roleplayers who don't like OOC communication at all. It disturbs their sense of immersion. But, if you're new in town, reaching a hand out in an OOC tell is never a bad idea. Most roleplayers do want to see more new roleplayers and, if you're reasonably well-spoken, they'll point you in the right direction.
You can pose again. After coming in you'll need to find a spot or do something like getting a drink. What kind of seat a character choses, and how he or she sits, can say alot about them. What kind of drink does your character like? How does your character drink? Delicate sips, big slurps...with a curly straw? Maybe she doesn't drink at all. Maybe she smokes (what do they smoke here? Pipeweed, hashish, cigarettes in a holder or maybe a fat stogie?). Or maybe he's there for something else - looking for someone, to exchange information with the bartender? But as before, don't overdo it. Your job as a player at this point is just to establish that you can write fairly well and aren't an attention hog.
You might want to get involved in someone else's conversation. If you're feeling lucky just sidle up and add a bit of commentary. Otherwise, you might ask in an OOC /tell if you can join them.
Now one caveat is that bars, taverns and cantinas don't always have the most scintillating roleplay. Very often people are doing in bars what most people do in bars. Make small talk and flirt. Still, if you haven't found what you're looking for in terms of thematic roleplaying groups in the forums this is the alternate route. Often good roleplayers also come to these places to unwind from adventuring, to goof off, or even to talent scout.
Don't give up hope. Occasionally RP randomly when in public places. Interact with NPCs. Write a brief pose about saddling a horse. You never know who might be passing by.