I listen to the song over and over to get a feel for the grooves/fills, and then just practice each one individually until I get it. A lot of the time I add in my own stuff as well, or if I can't figure out what the drummer is doing, just improvise.
I dont know if youve ever played guitar hero, but when you go into practice you can pratice the song in sections. Thats what you should do when trying to learn a cover once you get one section down move on to the other then combine them.
What I do is listen to the songs while driving, that way I get (unconsciously?) familiarized with the structure of the songs. After that, I only review the songs and polish them by looking out for the smaller details and fills... and voilá.
I'm not sure if that's the best way to learn covers, but it does work for me.
I usually get the "basics" down first...there is always that foundation rhythm and beat.And then go from there.If you plan on playing in a cover band,you'll find that people(regular,not musicians) that are listening are just into "hearing" that tune rather than...WOW! That drummer tore it up! Its usually personal preferences that drive US,including me to jazz it up more than the original.Get out there and experiment...you'll see!
I agree with what many others have mentioned, just listen to it a few times to get a feel for the song, then play along. When I play along with my Ipod though, I always leave one ear open so I can still hear my kit well. I don't want to drown out my own kit to think I'm playing the other part well, but then when the time comes to play it on my own have some parts sound sloppy.