The more you immerse yourself in each style (listening to the music, learning drum parts, learning the nuances/vocabularies/feels of each genre), the better you will understand and be able to play them. That's the big secret. Oh, and playing in bands of said genres will be the final step...
1. Get lessons from teachers who are well versed in the styles you're interested in.
2. Immerse yourself in the various forms of music that interest you, try to identify what fundamentals are involved in each style and aim to learn these. Look/listen to what the drummers are playing. What are the fundamental elements of each style? ie. rock drumming relies heavily on a solid 4/4. Big band drumming requires you to learn how to swing, extreme metal requires good command of your feet etc. (Very broad here I know, but an understanding of the genre is important)
3. Buy books/dvd's that are genre specific, i.e latin grooves, african rythms, metal drumming etc.....whatever......as long as there's a basic illustration of the fundamentals you're trying to learn.
Go see music live! Go to somewhere that has a live jazz band or some rock band. Its nice to experience music where you cant hit the skip button. I like pandora radio also...type in genres or an artist to hear a nice range of music.
All of the advice in here is great, and true. If you want to know a style, study that style. Read it, listen to it, find a teacher for it, etc, etc, etc. Then, after you get decent at it, practice it from time to time, even if you don't use it regularly. That's what I do, and I am well versed in a lot of styles from it.
Imitation is the sincerest form of flattery. Eventually, once you've listened to enough of the genre you're trying to emulate, and spent enough time working on emulating the finer nuances, you'll cease to be imitating.
While enough can't be said about immersing yourself in a particular style to learn it, I'd say a great resource for surveying many styles is Tommy Igoe's Groove Essentials. Especially the play-along book.
One thing that really helped me get versatile is putting myself in playing situations and gigs where there was a wide variety of styles. I've played in three different praise bands at church (all three of which have radically differing styles), two musicals, and several sessions/gigs in the local area that cover everything from rock to country to swing. Learning the parts is a challenge and presenting the music authentically is key in these situations. I have found that accepting these "out of the comfort zone" gigs has really helped me grow as a drummer.
Maybe going out and getting gigs is not where you are at right now, but definitely play with a lot of different musicians and bands to expose yourself to different genres. Currently, I'm planning to take jazz and latin lessons when I retire from my current job and have some time on my hands.
I suggest you expose yourself to different styles of music by listening, watching videos and practicing the style. Then let the choise of style happen naturally. You will most probably be natural in 2-3 styles and accomplished in some other styles you expose yourself to. I think the most important thing to do in learning a new style is listening to it as much as possible.
my drum teacher recommends that when you're trying to learn a new style or genre you should try to learn from the most authentic sources possible. for example, when learning afro-cuban, study the latin music coming out of cuba and new york in the 30's and 40's. move forward in time to newer music but always try to find the most authentic sources you can. what you end up playing for yourself may not be exactly what the original musicians played but at least you know your influences are authentic.