1-Delay and/or Don't Go
This isn't going to seem like a great answer, but one idea to consider is to just delay the appointment and/or just stop doing that appointment. Keep in mind that the time you are "stuck" home is really quite short in the grand scheme of things. I know it seems like forever, but I promise it goes by really quickly. The minutes crawl, but the years fly. It won't be too long before either your youngest is old enough to come along or your oldest is old enough to watch siblings.
You might have to get your nails done less frequently--or maybe even stop for a year or two. You might be a month or two late for the dentist. This isn't all ideal, but sometimes it is life with young children, especially a newborn. I know that with each of my kids, my dentist trips during the baby's first year are more than 6 months apart. It isn't what I consciously plan, but it happens. The day will come when you can leave your child home more easily so you can go to your appointments.
2-Get a Babysitter
The obvious here is to get a babysitter so you can go to your appointment. If you want or need to go and you can't or don't want to take your child(ren) along with you, then you need a babysitter. This could be a young babysitter you pay. It isn't really ideal because a) who wants to us babysitting budget money for a dentist trip? and b) it makes the whole thing even more expensive!
It could also be a family member (see my post with tips for using family as babysitters). I primarily use my parents for watching my kids when I have an appointment to go to. If you don't have family nearby, you might have an older neighbor or friend from church who would watch your children for you.
Another great idea is to trade with a nearby neighbor or friend who has small children, too. You watch her kids while she goes to the dentist and she watches yours while you go.
You could also get a neighbor to watch and schedule during naptime. I know my kids are easiest during their naps, so for someone to either have my child sleep at their house or sit at my house while my child sleeps, it is easy. I know a lot of people who live in apartment buildings or close homes will leave a monitor with the neighbor and a key to get in if the baby needs something.
3-Use Your Spouse
For dentist trips, typically my husband stays home and I go in early for an exam and then I drive home and he goes in for his exam and then heads to work. He can then work through lunch to make up for the late arrival if needed. We make sure to tell the receptionist when we are making the appointments what we will be doing so they know that if they are running behind, it will make the spouse late.
You could use the same idea and have your spouse either watch the kids in the waiting room or take them to a nearby park or run some errands nearby while you do your appointment. Perhaps your spouse works somewhere that allows children to visit and your spouse could even have the kids at work while you did what you needed to.
When my husband worked too far away to do a swap and still work, he just took a day off. He needed to see the dentist, so he had to use PTO to do it. Then I did my visit the same day. If you plan in advance, you can make lots of appointments all on the same day and take time off. It is just part of the deal in being a parent with young children.
Some places might even be open on weekends and you can get things done while your spouse is home from work.
4-Find a Kid-Friendly Location
Some places are set up to be able to have children around. If you have places like these, use them.
Another point I want to make here is to do take your kids when it is reasonable to do so. For my OB appointments when I am pregnant, when I had a regular visit, I took my kids. That way I wasn't over-using the resources I had for watching my children. My OB has a basket of toys in each exam room, so it is very kid friendly.
Some places might even offer childcare on sight.
5-Schedule While Kids are Busy
As your children get older, they will likely start to get involved in things. I often make appointments while my children are in school so that when I am asking someone to watch the remaining children, it isn't so many.
6-Think Outside the Box
When it comes to a doctor or dentist, you kind of have to do things the conventional way or going to them and having your exam done. For other things, however, there are a lot of options. I have a good friend cut my hair instead of going to a salon. She even comes to my house to do it. She comes during naptime and I am able to get my hair cut without Brinley being an active participant. Since Brinley has been old enough to get her hair cut, we schedule it for just before nap. Then she gets her hair cut, I put her down, and then we do my cut.
For a short time, I enjoyed getting my eyebrows waxed. I find that to be something I don't need to regularly spend my time or money on, so I have been using at-home waxing kits for years. So far, I haven't done any damage to the eye-brows :).
Hopefully you will find something here that will help you have an easier time working in your appointments. Do you have other ideas that you utilize for getting your appointments in?