I play church gigs, where most weeks, I'm learning a new song or two, and learning new arrangements, adapting to a different key on a song or two, etc.
If it's a new song, I sometimes (happened this week) learn it just by playing along. For more complicated stuff, I'll make a chord chart - stopping the tune as I go along to write everything down. Sometimes, we have charts provided
After I have a chart, I play through it a few times, then, once I know a certain part of it, I start looking away. As I play through it more and more, I look away more, and I eventually can look away for the whole thing. It takes a couple sessions usually to get something committed to memory, and after that, it's repetition to polish things - get everything nailed to where the chord structure is in my head. That way, when I get to the gig, I can adapt the part I'm playing to fit what's happening around me - the actual part I play is something that I'm working on through rehearsal, run through, and when we do 2 services, I'm usually still tweaking a bit here and there for the second service. The key is that the chord structure is firmly embedded in my head, so I'm just working on the particulars of the part I'm playing at the gig.