When starting from lyrics, I think about what kind of feel the lyrics suggest. For example, I might think: "okay, I know I want a minor key for this" or I might think "I want a more complex or changing time signature for this; I want it to feel a bit nervous or uncertain..."
From there, I might try several things. I'll often sing the lyrics a couple a different ways, without any accompaniment, just to see where that leads. If I come up with something I like, I'll find a good key to match the melody line or chorus or whatever I've sung. I find singing to create the melody (or whatever) is often a good approach, for me. That said, sometimes I'll write a melody on an instrument. For some reason, that instrument is often piano or a block flute (even if I don't use the instrument for the actual song), but sometimes it's guitar.
If I'm starting from a time signature, I'll usually come up with a drum beat or a riff or rhythm pattern. For example, I'm working on a song right named "Scrupulous" right now. The lyrics come from a pantoum poem I wrote years ago. I wanted a somewhat disjointed rhythm, and I wrote a guitar riff that is structured in beats of 5, then 4, then 3 (making it kinda a form of 12/8). Then I created drums to go with that. Then I sang the lyrics over it to work out the melody line. Then I added bass.
I don't think there's a single approach. Try different things and see what works for you, and for that specific song. You'll probably find that songs will develop in different ways and using different approaches. The key is to keep working at it. Don't wait for inspiration. Let the process of working at it birth inspiration.