Assuming we're talking about a world similar to ours, the only difference being the existence of mermaids. They are fish with an upper half visibly similar to a human's, who live in the sea, can survive at any depth, can't breathe air, and have an intelligence and culture as developed as a human's (though their culture isn't necessarily similar to ours). What would be plausible modes of transportation for them, for both short and long distances?
Let's start low-tech and work out way up. There's always animal riding. Some types of fish, or maybe dolphins or squids, could be tamed or domesticated and then ridden around. You can then have them pulling vehicles along, like horse-drawn carriages. The first vehicle not using animals could be something similar to a sailboat that uses ocean currents. The upside of using currents is that the big ones are reliable and always present. The downside is that they are often one way in a very big loop, making it slower to sail in the opposite direction and infeasible for short or medium-distance trips to go all the way around.
The next step would be self-powered vehicles. The first ones would probably have more in common with aeroplanes than with cars, since there's so much water to push against and keeping off the sea floor is pretty easy. A propeller works fine, the problem is what powers it - basically, building an engine under water. It probably pretty doable though, as long as you can build contained systems. You could also trap some air in the top parts of your vehicle and make surface sail boats, that can go in any direction like the sub-planes but can move without engines like the current-riders.
Closer to modern age, they could also have conventional air planes. The biggest difference is that the planes need to be watertight and filled with water, and therefore much heavier. You can still do it, it's just more expensive than our plane flights - probably only worth it for quick travels across landmasses. If you need to get across a landmass and is not in as big a hurry, or don't have as much money, you can enlist help from the humans and their railroads or highways. Have them make train cars or buses that carry mermaids, step on in one coastal city and off in another. This could also work when moving along a coast, depending on relative costs and speeds of railways and conventional underwater transportation.
1$\begingroup$ Sailboats are not "often one way". It is possible to travel to a point in any direction relative to the wind, though the speed you can travel varies. $\endgroup$– user16107Aug 9, 2016 at 10:36
$\begingroup$ I meant that ocean currents are one way, and this affects your speed in the other direction. I'll edit so that it's clearer. $\endgroup$– GrolloAug 9, 2016 at 10:40
$\begingroup$ you could build a sailboat that does not have to be worked from the air side, $\endgroup$– JohnOct 8, 2020 at 13:55
For short to medium distances, they might use small underwater propulsion devices like those used by divers.
They probably wouldn't use large, enclosed underwater vehicles like submarines. The cost in drag for a large enclosed vehicle underwater is too high. A submarine only makes sense because of 1) stealth and 2) the need to provide an air-filled environment for humans.
For longer distances, it would make most sense for mermaids to travel above water, just like humans: in boats, airplanes, and even (possibly) ground vehicles. They would need a breathing device or even a tank to sit in, but it would be worth it in order to travel more efficiently through air, rather than water.