When I was attending school at the Recording Institute of Detroit, I was taught when mixing that you should start with the kick drum first then move to the bass and so on. After working on mixes and conversing with other veteran recording engineers I found out that although it might be the easiest way it is not the best way.
TIP: Never assume the way you were taught or the way you taught yourself to mix is always the best way. Each project should be mixed according to it's individual needs.
What a recording engineer should do is think about what the vocal point of the song is and start with that. If the vocal point of the song is the vocals then you should start with that. If the song is more geared around the lead guitar then you should start with that.
Now a lot of people may say that if that is the case then you would probably start with the vocals all the time, but this is not necessarily always the truth. A perfect example of this would be rap or hip hop music in the last five or ten years. The cleverness of the lyrics from song to song have really diminished. In fact, in some cases fresh new lyrics have almost disappeared all together. If the lyrics in a song are just not that explosive, there not going to be the main thing that is going to make the song sell. There could be cases where the courses or the hook of the song along with the baseline is what sells, and the verses, well, just aren't that much to talk about. Mix the explosive hook and baseline line out front, it's what you want the people to hear, then when the verses come along focus on maybe making the groove something that is more of the focus point. It's can be difficult but it's possible.
If you know the artist well enough, you might be able to make some suggestion like shortening up the verses or changing them. But, in any event the ideal hear is to make it sound good all the way through the song, especially if your name will go on it as the engineer.
Other cases like mixing progressive rock/metal or techno, you might find yourself in the same situation. You might have to listen to the whole song and throughout the song you may need to mix differently depending on what you want the audience to be focused on the most.