Do you need an instrument to write a song? Well, I think it helps. You can predict how the song will sound with the music, a lot better than maybe, humming it in your head. One of the reasons I brought a guitar was so that I could play the songs I was humming in my head. As well as guitar, a piano is a great instrument to use when writing a song. You can map out melodies better with the keys, as well as playing the chordal patterns.
Should you write the lyrics first, or the music? This is often something I contradict myself with, and it all depends on the songwriter. When bands write songs together they usually split the music and the lyrics between them. My concern is that sometimes, when you have written the music and lyrics separately, you can tell. The words sound very broken and sometimes rushed, because the singer is trying to fit the lyrics and syllables into the music.
I find the best songs I write come from jamming on a guitar or playing on a piano and singing along. Singing anything that sounds good to it. Even gibberish. Once I have the structure of the song, I then start putting meaningful lyrics in place of the gibberish. Sometimes I even keep some of the gibberish. Take the band Sigur Ros. If you have never heard of them, I suggest you legally download some of their music now. They believe the vocals are an instrument, and they treat it that way. The singer sings utter gibberish, but it sounds good. It sounds totally improvised and natural.
Another good way to get ideas for a song is to start jamming with others. Think of a little riff. It could be something at random. Get the other band members to join in with something that fits in.
Lyrics wise, I think you should do what I said. Use gibberish at first to work out the melody and syllables, then when you have finished the structure of the song, write around the gibberish!
There's a lot of software available that aids in writing songs like Garage Band for the Mac, and Cubase for Windows.
