I’ve never hired a babysitter I didn’t know personally (as someone’s child, usually), but I did do a lot of babysitting, both as an amateur teen and as a professional adult. My initial calls were mostly word of mouth, but here’s what kept people calling me back:
Always be on time. Get there 10 minutes early and sit in your car so you can press the doorbell at the stroke of On Time. Don’t be early ringing the bell - Mom is probably still getting ready and you’re going to mess up her rhythm - but don’t be late, either.
Bring child entertaining things with you. I carried a large rolling bag full of coloring books and puzzles and crafts and play scarves and other activities. This works two ways - one, someone else’s coloring book is more interesting than the same old boring coloring book a kid has had since last Easter, so the kids loved it and it kept them busy and out of trouble. Secondly, it showed the parent that I intended to be actively involved with edumacationalish time with their kids while they were gone.
Ask the parent when a good time would be for the kiddo to call them for a check-in. Then set an alarm in your phone for that time. Again, this does several things: it lets the parent know that you know kids need their folks. It lets the kid know that Mom will be reachable, and staves off quite a bit of bitching and moaning in the meantime. It lets you look all responsible and awesome when you’re on time with the check-in call. It eases Mom Guilt that she’s abandoning her sprog.
Speaking of Mom Guilt…there will come a time when the kid does some First while you’re on duty. Don’t breathe a word of it. First steps, first word, first anything…just keep your mouth shut. The momentary glee of sharing their accomplishment will be far overshadowed by the Mom Guilt that they weren’t there to witness it. Just don’t go there. This is one time when ignorance is bliss.
Clean up after yourself and the kids. If you’re going to be doing more than that, negotiate housework separately from childcare, in terms of time and salary. I did have one house I did clean the kitchen, vacuum and make beds for, but they paid me extra for it. But I wouldn’t charge extra for washing the dishes that the kids and I dirtied.
Be prepared to bring your own food. Most people will tell you to “help yourself to anything,” but they don’t always mean it. OTOH, some people will be offended if you don’t eat a thing of theirs, so this is a minefield of cultural expectations and personal quirks. I tended to bring a yogurt and a small container of almonds or something that would get me through the shift, but if they left a pizza for the kids and there was extra, I’d go ahead and have a piece of that, too.
Good luck! Get yourself registered with a website like sittercity or nanny.com, as well as any locally based sitter services you like. I got only about a quarter of my gigs from such places, but I bet it would be more now than it was 10 years ago.