I pretty much listen for everything the same. If the singer doesn't have a somewhat good voice or if the lyrics are really dumb then i won't listen to it. The music doesn't make too much of a difference unless it's like country music with moaning violins in the background or really gay tremolo'd pinch harmonics everywhere.
When it comes to hip-hop or rap, I'm pretty much focused on the lyrics but when it comes to other genres like Rock, Alternative, Punk, or Pop the importance of words fades away. For instance, Blur Song 2 had some terrible lyrics that made no sense to me, "...head checked by a jumbo jet...Wahoo!" but I liked the song for the opening drums. Then there is I'm yours by Jason Mraz where the lyrics and instruments are both great.