Obviously the original recording is going to play a huge part in the final product, but that doesn't mean you can't do a lot to have a great music listening experience.
What kind of equipment to buy is more subtle than "x headphone is god for y genre." You need to consider personal taste in you selection. For example, I'm very picky about my bass response. I love bass, I think it's a vital aspect to music, but by the same token, it shouldn't be overwhelming. So, what I consider too bassy even for genre's like EDM are just right for other people.
Granted, there are going to be aspects to genre's that are going to play a big part. I'm also pretty into metal (I tend to like melodic death metal and thrash metal) so a lot of really fast, precise guitar and keyboard parts come up a lot. I find that headphones such as the Sennheiser 600 (and the 800 for that matter) were quite sloppy with the lead instruments. Of course, for headphones like that, amping is going to make a difference.
Currently I run an audio-gd compass -> AKG K702s. I really enjoy them for all kinds of music, particularly metal and classical. They are beautifully detailed and clear. I think they do a fantastic job with the details with vocals, which with metal is going to be a bigger consideration than some other genres.
In terms of file quality, I think it's good practice to rip to lossless. Even if you can't hear the difference in this rig, or the next one, or even the next one, if you decide to go down the path of head-fi (sorry about your wallet by the way) then it's likely that you will one day have a rig that you can tell the difference on.
So, at the end of the day, make a CD/bring an ipod/computer/flash drive etc with high quality versions (preferably lossless) to a meet or two, see if there's a store that sells hi-fi headphones in your area, and play around a bit until you find something you like.
Oh, and the plus side to listening to crappy headphones is your new ones will sound that much better by comparison.
good luck!