I can only speak for myself, of course, on the way that songs "arrive" but for me they generally start with a phrase - music and words both arriving together. This might be a fragment of a chorus or a verse but it usually comes to me when I'm playing guitar. With "I Wish I Was" it was the result of me playing around with James Taylor's "backwards" version of the D major chord. "(Lighter Than a) Butterfly's Wing" came after I was experimenting with Paul McCartney's "Blackbird" chords. Exceptionally, my new song "Fallen on my Feet" came when I picked up a borrowed ukulele and started strumming it.
The next stage, which usually follows immediately, is to extend and develop that phrase using techniques which have been around for centuries. Generally if a musical phrase in a verse tends to go up or down I'll try and develop that tendency and then save the contrasting motion for the chorus. There's an unwritten rule in writing melodies that you can get away with 2 repeats of a phrase (3 altogether) but by the third you'll have lost the listener unless a) your name is John Lennon or b) you've just written a great lyric called "Imagine" :)
Writing additional verses is usually the hard slog of song writing. I try to keep the same lyric rhythm patterns from verse to verse and if I have a hook line (i.e. one that hooks and draws the listener in) then I'll often repeat that line exactly in every verse.
The bottom line is: do the lyrics sing well? If not then they have no business being in the song. I'll often sacrifice overall "meaning" of lyrics if they sing well and trip off the tongue. ("Hey Jude"'s "The movement you need is on your shoulder" sings beautifully even though Macca was unsure of it until John Lennon approved the line.)
Once the song is written I'll practise it until it feels comfortable and usually somewhere along the way I'll type the score of it into Sibelius (score-writing program) so that I don't forget nuances of melody etc. I'll also use Sibelius as a scratch-pad to play with arrangements and instrumentation.
And then the next stage is to import any MIDI tracks I want to use from Sibelius into ProTools and so begins the recording process.