I was in a similar situation where I joined a cover band and had to learn roughly 80 songs. Took me a few months to be honest but here's the system I developed:
- Open a document editor
- Copy/paste the song lyrics into two columns
- Listen to the song, and notate at the top the feel of the beat
- As you listen, mark what beat you want each section to be - find your own shorthand (e.g. 2/4 beat, surf rock, disco, etc).
- Also be sure to notate where you fill, and transitional phrases from part to part (e.g. 8 bar transition, 12 bar breakdown).
I have the lyrics in black and then my personal notation in bright blue so I can differentiate the two. Hope that helps!