How can I determine the approximate time required to fly a particular plane (e.g. a Cessna 172) between two points, considering the possibility of needing intermediate stops? For airline flights, I can just look it up online; is there something similar for GA? I'm considering getting a license and plane at some point in the future, and there are two cities quite far apart that I'll be traveling between fairly frequently, and I would be interested in seeing if it would be feasible to make that trip in a plane I could possibly afford at some point.
No, there's no such service for GA pilots. In fact, there's no service for commercial planes, either; the only way you know ahead of time how long a particular flight will last is because someone at the airline calculated it and then published their numbers. Unfortunately, as a private pilot, you don't have a company to do your math for you, so you have to do it yourself. (There are apps, such as Garmin Pilot or ForeFlight, that will do it for you, but they aren't free.)
To help your decision whether or not to get your license, airplane manufacturers will publish the typical effective range for their planes. You just need to get a map and a ruler, measure the distance between the airports you plan to fly to, and see if one number is less than the other.
The time required is much the same. Divide distance measured by published cruise speed to get time. This won't be precisely accurate (your plane doesn't fly as fast when climbing, you might have to navigate around obstacles, etc.), but it will at least get you in the ballpark.
If you do decide to get your license, part of the classroom training will be on flight planning. You'll learn how to use a map and a winds aloft table to calculate your exact flight time, fuel usage, etc. for any given flight. And part of that is identifying airports near your flight path that could be used if you wind up burning more gas than you calculated and have to stop for a fill-up.
Range in miles isn’t perfectly reliable since it depends on winds aloft, which vary by hour and by altitude. If you find a single number for a given plane, be aware that it does not account for this.
More commonly you will see range given in hours. If not, you can calculate it from the size of the fuel tanks divided by the fuel consumption per hour and then subtract an hour for reserves. You can get no-wind range in miles by multiplying that by the cruise speed.
While that number isn’t good enough for real flight planning, it’s good enough to get a general idea of whether a trip is practical, or roughly how many fuel stops you’ll need.
In the example you gave in a comment of 2,200nm in a C172, we can guess that would take about 20 hours. For a rare event where the trip itself is the point, such as a fly-in like EAA Airventure in Oshkosh, WI, that might be reasonable.
For something you’ll be doing “fairly frequently”, though, that seems ridiculous. You’d want to look at a significantly faster plane, which will take you time (and money) to work up to.
Also, keep in mind that GA is rarely cheaper or faster than airline travel, nor as safe or reliable. So unless one of your points doesn’t have airline service, this plan doesn’t seem remotely realistic.