I saw them entirely accidentally. I was looking at the moon through bins and checked out The Other Bright Thing nearby and was shocked to see what looks like little dots next to the big one. As Galileo himself would have done, I immediately went on the internet and consulted a sky chart, which confirmed that I was seeing the moons of Jupiter.

I still occasionally drag my friends out to look at the moons on a clear night. It's my favorite bit of practical astronomy to share.