One of the inspirations for this was Tactics Ogre: Knight of Lodis, where I liked their depiction of slithering, combat-worthy mermaids.
I've been trying to develop mermaids who make pilgrimages to land cities, and have been considering mermaids with shore-side villages and towns. For this purpose, I've been trying to work out ways they can be more capable on land (not as capable as humans, but more so than a flapping fish).
For modes of movement I've considered... The main one would seem to be shuffling/crawling, since they likely can't slither. They could assist this motion with their hands, with their tail, and potentially with extra fins near the waist (the picture below has an interesting example of an elbow-like fin). The main issue with this is if the mermaid is heavy. If they are light and flexible enough, skillful manipulation of these appendages might make them more able on land than seals. Trying to scoot on the floor is pretty painful,but a mermaid's tail should be tough enough to endure this (though long-term travel over hard ground, they may get chafed bloody). Rolling and springing up with a large jump also seem plausible short distance movement.
EDIT: Forgot to mention inch-worming. If the mermaid is quite flexible, then this may be a possibility for small movements and adjustments, as well as assistance in the crawl/shuffle.
Mermaids may need to bring salt water and sponges with them when on land, to dose themselves with. Unless they can convert to fresh water creatures quickly, then freshwater would likely be unpleasant to them. However, they could probably bring enough salt around to make fresh water usable for a sponge bath. Drysuits filled with saltwater may also become popular.
Human forms of transport, carriages and cars would work fine for mermaids if they can arrange them. Some sort of mermaid-bicycle using the hands or tail would be interesting to see... though wheel chairs may be the norm.
My question is to do with how to make the locomotion and daily life of mermaids on land plausible and feasible. Notably, you can expect they'll spend a lot of their time in the sea, even with their coastal villages/towns (pilgrims will have more trouble).
I also put some thought into mermaid land combat, but I'll save that for another day.