Babysitting without your mom around is pretty much the same as with her around, only you have some more responsibilities with it. Just treat the little girl the same way you do when you're with your mom, but make sure she treats you like an adult because you're the one in charge then, not your mom. Keep all phone numbers to her parents accessible, and don't be shy to call them if it's a serious question you need to ask. If you're babysitting at their house, dont' be shy to ask questions, such as if you need to feed the girl, where the food is, how to operate the oven or microwave if you're not used to it, what channels they're allowed to watch on tv, etc. Over all, just make sure you're focused on the girl, not yourself. As long as you keep a level head, you'll be just fine and won't have a problem at all!

As to this new girl, just go ahead and tell her! Be nice back, try to find mutual interests, make the effort to talk to her and interact. Find ways to just hang out, and see if she's worth making friends with. Don't be shy or hesitant, just go and talk and say flat out you wanna be friends!

I hope both go well for you!

Joss, I don't really have an advice for you. I've never babysat (Though I have been a CIT at a camp) and I don't really want to. Also, I don't have any friend-making skills at all, so I can't help you with that. But I do hope that all goes well, and that you make friends with the new girl!