Always starts by jamming out a guitar riff or bass line, usually to a drum loop. Then if it began with guitar, I'll add bass. If bass, I'll add guitar. Maybe add a keyboard part. Somewhere along the line I'll have written out the chord progression. Then I'll loop the whole thing and play it back repeatedly while trying out vocal melody ideas, singing inspired things like "oooh, oooh, oooh," or "yadda, yadda, yaah!" In short, making a total fool of myself. So I do it late at night when I'm the only one awake. If I'm liking the melody, I'll keep going with it. If not, it joins a million never-to-be finished song fragments on my hard drive. If I keep going, I'll decide whether this part is a verse, chorus, bridge, etc., and try to create other parts that go with it by the same process.
Somewhere along the line, those "oooh, oooh, ooohs" will have become words and phrases, which will trigger associations in my mind leading to other words phrases. If not, I'll abandon it and come back later--maybe months later. Or maybe never. As I begin to work on lyrics seriously, I'll spend some time asking myself what this song is actually going to be about--who is speaking, to whom, about what? I'm really not sure how or why the lyrics come, when they come. Lyrics are by far the hardest part of the process.
One thing I have forced myself to stop doing is layering on more and more instrumental tracks, tracking and retracking, on a song that lacks vocal melody and lyrics. I used to waste a lot of time that way. I'll draw a line and tell myself: "I'm not doing another damn thing with this until I have a vocal." If I'm making progress on melody and lyrics, it's potentially a song. I'll keep working on it. Otherwise, it's an unproductive waste of my time. I'll move on to something else.
From start to finish, a new song tends to take me two or three weeks, but sometimes much longer. I'm usually working on three or four new songs at any given time. I'll rotate between them depending on what is exciting me at the moment. When I start to see the light at the end of the tunnel on one of them, it will get my undivided attention until it is finished.
I need to set aside a few hours every week to jam and record the results in order to keep the queue of potential song ideas full. That's my process.