I would just have them come to my home to babysit. If you take them to someone's home then they are in fact running an unlicensed child care business out of their home. That is not legal in some states. Some states do allow a person to watch one family's children without a license but I would still not want to do that.
Having this person come to your home gives you total control of the situation. You know there are no animals, you know what they are doing because they are doing it in your home.
I would have them watch the child while I was home a few times to see how it goes, how they interact, how they discipline them if they do something against your rules...you know, do they say NO loudly, do they say come over here sweetie, let's do this instead, do they just pick them up and move them away....
Either that of find a Mother's Day Out program that is open a few hours per day a few days per week. They usually have you sign up for at least one regular day per week where your child comes. You don't have to sign them up for the entire time. If you miss you should not have to pay anything either.
Doing this would allow you to also get to know other moms in your new town. They use this program too and your little one will be forming future friends and play date companions.
The teachers at most MDO programs are experienced mom's themselves and might be your future babysitters. That was always a great thing for me since I knew them and my grand kids knew them and were comfortable with them.
knowing in advance that my grand kids would be watched that one day per week was a great way for me to arrange those wonderful doc appointments that I really didn't want the kiddo's to attend. When making a doc's appointment you can always tell them you are available between the hours of Xam and Xpm on Xday of the week.