I am fortunate to be looking after two very spirited, very different little girls. One, C, is 4 1/2 and the other, S, is 2. I am also fortunate in the fact that their parents want them to get the most out of their childhood and the city in which they live. Both girls take various classes: C has soccer, gymnastics, swimming, and piano. S takes music, cooking and goes to a play group at the family's church.
If the kids you watch aren't enrolled in any such classes, it may be more challenging for you to meet new people. Do some research--a simple google seach should suffice- there are loads of places that offer classes of all sorts that you are able to "drop-in" on. Some of course, require registration, but I have found plenty of facilities which work on this basis in order to accommodate families.
Talk to your boss- Chances are if you find something you think the child (or children) will enjoy, they will be willing to support it. Sometimes it may be under $20 a week to do a drop-in. Try a few to find out what they like---they may still be figuring it out!
Taking a class with your kids will benefit you both. If you're a new nanny, you need an opportunity to bond outside of the home. If you're an old hat, it may help to keep things exciting and fresh. Just like any relationship, you need to provide arenas for you both to grow.
If your kid is a little older, there may be many classes which are drop-offs... This could still be a good thing. Giving them a place to go that's a little different will build confidence. It will also build their trust in you!
Finding a place to take a class that you both enjoy will help you meet new people which can turn into lasting friendships!