Question: I would like to know when is a good time to start teaching your baby sign language. I heard they won't start signing back to you before 6-7 months so starting before 3 months might be pointless. Any thoughts on when you think would be a good time to start for your sake as well as your baby's?

Answer: 
Just do it as normal all the time.

They pick up everything all the time.

That is how children learn by seeing, hearing and experiencing things.
We started toilet training our child from roughly day 5, almost by accident. By month 2 or 3, it became clear that the lack of communication was the biggest impediment to success (the baby can't take off their own diaper or climb onto the potty) and it took my dumb brain a few months to come up with sign language. We (the parents) use about 10 signs (of the 25 plus letters that we know - we're very green) as we talk and the baby is currently signing 'milk' to communicate a need (maybe milk). Their verbal communication is still likely months away from catching up based on what I've read. The gross motor stuff is just easier to learn than coordinating the fine motor, breathing and vocalization needed for talking.

The baby made their first poop joke at about 5 months (it wasn't very good, but 5 months!) and it seems clear that babies have a lot going on that they can't express. Right now, our limited vocabulary feels like a missed opportunity.