Late afternoon or early a.m. appointments, so husband or significant other can work from home or come home early --that's what we did when our child was young.
I know it's not what you want to hear, but frankly, many doctors do not allow young children to tag along on visits, period. I even had one ob/gyn -- yes, she delivers babies! -- who did not permit young children in the office at all. You are eventually going to hit this issue so I'd work now on finding someone who can do appointment babysitting.
Also, some offices do have Saturday hours -- not many, but some like our general internal medicine doctors have Saturday hours until early afternoon. Seek out doctors who have those hours if you possibly can. If they don't publicize Saturday hours, ask -- some offices have them but don't like to make it widely known!
If you don't have a husband or SO or nearby, trusty relative in the picture, and rely heavily on the teen babysitters, see if there is a college or university nearby; some college kids babysit and they have MUCH more flexible hours. There are often online sites where you can find college age kids who do babysitting. It would be good to establish a relationship with a couple of reliable older sitters now, before you need one for the next doctor's appointment. I notice you're in Blacksburg -- you have a large university right there and if you search online you might find a babysitter board where college kids offer their services.