The songwriting process is different every time, but it usually goes down some familiar roads. For example, I usually start by writing music first, or writing music and lyric at the same time. I may suddenly come up with a phrase and a melody for that phrase (which is how I came up with the song 'A Southbound Train'). But the music almost always precedes the lyric in some way.
Trying to write lyrics with no music is difficult because lyrics I write sans musique tend to follow some simple meter, whereas melodies I come up with may not match that meter at all. While I have several lyrics with no music, they almost never get turned into actual songs. I dislike feeling constrained by the meter of pre-existing lyrics.
I try to write melodies instead of writing chord progressions. It's easy to mess around and come up with a cool chord progression, but often it doesn't work out naturally to having a melody. I tend to overcomplicate myself when writing chord progressions anyway, so I find it's better to start with melodies and then consider the accompanying chords.
Another important thing to consider - will this be a vocal piece or an instrumental piece? Some melodies aren't really straightforward to sing, or cover a very large range. Sometimes I want a particular instrumental palette instead of the voice. And sometimes I think it just works better instrumental than vocal. 'The March Song' was originally meant to be a vocal piece and it had lyrics written, but I decided I preferred it as an instrumental.
Before I can really begin the production process (recording, arranging, mixing), I usually need to have the lyrics finalized. This is so I can put down a vocal track and know that the song is in the correct key. Having a good vocal track is also inspiring for the production! I often sing dummy lyrics as I write a melody, so if something from those dummy lyrics resonates with me, I try to branch off that. That's how one of my songs, 'Sitting by the Sea,' came to be. If absolutely nothing comes to mind, it's a real challenge and I probably need to sit with it and work it out.
The form is another important thing to consider. I usually do verse-chorus (ABAB) with an optional bridge, and I like having pre-choruses too. I appreciate that verse-chorus gives you a structure and a direction to go with a song while also giving you a fair amount of flexibility on how to do that (have a pre-chorus! post-chorus! cut the chorus halfway! start with a chorus! bridge, instrumental solo...). Sometimes, in an old timey mood, I'll do AABA. I basically never do strophic form (AAA...), though I'm considering trying it out. Recently I've been challenging myself with forms beyond AABA and verse-chorus. 'Sitting by the Sea' was an attempt at arch form (ABCBA) though it really came out as (ABBCBD). A recent song has even become quite strange, with a near through-composed middle part but sandwiched by a repeated chorus... Form can be an easy way to spice things up, but you have to be careful with it. Listeners like repetition because unless your melody is an immediate earworm, it can be easy to forget. Not repeating your melody allows you to have longer phrases and more varied textures, but you need to be careful not to lose your listeners. When you stray from established forms, you also need to do more 'work' - what is this middle bit doing? Should I add more material or am I pushing it too far? Verse-chorus has already done the work for you! Verse-chorus also lets you be very economical with your material - write a few good melodies, rinse and repeat. You need to write more melodies when you venture outside traditional song form, so it's more work. The reward is that a different form lets you try different things - you can elaborate a melody in a way you don't really have 'time' to do in Verse-chorus.
This previous section shows a recurring struggle for me - keep it simple, or experiment? I love experimenting, but things can get out of hand sometimes and the end result may not be listenable! At the same time, I don't want to be lazy and assume every song needs to be verse-chorus with a few melodic phrases. Finding this balance between experimentation and conventionality is one of the challenges of songwriting for me.