My problem with this is that it feels like my plants are not native to the ocean, but only transplanted from land.
Part of the problem here is that things that grow in the sea don't have to put up with nearly the same amount of inconvenience that things that grow on the land, do. The forebears of ancient terrestrial plants basically had to terraform the surface world, overcoming a whole bunch of problems that just aren't there for photosynthesizers that stay in the sea.
I won't elaborate on this here, but you don't really get stuff like roots, or fruit in purely aquatic plants and algae. This basically leaves you with few options, if you want to take your examples from real life.
Firstly, you have things where you eat the leaves (or fronds, when talking about algae rather than plants) or stems... obviously this has classic bits of human seaweed farming of stuff like kelp. Remember that algaculture is more than just the stuff that sushi nori gets wrapped in... there's scope for producing oils too, for eating or maybe fuel if your merpeeps spend much time on the surface and felt the urge to burn stuff. There are obviously lots of kinds of shapes and textures and flavour here, but they're mostly going to be "leafy-looking stuff" and "slimy stuff".
Secondly, you can, in fact, farm cereals. Seagrasses are one of many species that returned to the ocean having decided that the surface world wasn't all it was cracked up to be, and represent the only flowering sea plants. They're also actual plants as well, not algae. Tape seagrass seeds are edible, and can be cooked, though I'm not sure if it is possible to make flour from them (or if you'd want to). The grasses themselves can be used to produce fibres for making ropes and nets and maybe even clothing if you were in to that sort of thing. Seagrass meadows are also favoured habitats of Sirenia (mantees and their relatives) which might also be farmed for a change of flavour from fish, if your merpeeps were meat eaters. Clearly there's scope for farming both seeds and meat here. This might feel a bit too "transplanted from land" for your liking, but that's because these things are. Real life is not always exotic. Seagrass meadows can at least be made to feel quite different from grasses on land, with a bit of thought.
Thirdly you can harvest stuff that kinda just falls in the water. This seems a bit lazy, and isn't really farming, but more sort of lazy foraging. Coconuts are the most obvious thing here, and maybe they could be sorta farmed on coastlines, with a bit of effort, but it probably isn't worth it.
Relatedly though, you might consider things that are technically surface plants but are happy to grow in tidal water... mangroves. They form sometimes quite large coastal swamps which have all sorts of interesting and occasionally edible wildlife in them. Some species of mangrove have edible leaves and fruits... I've just discovered the mangrove apple but I don't doubt there are others. Mangroves drop their fruit into the sea, and may be planted in the sea, and so lend themselves to cultivation by beings who don't want to spend much, if any, time on the surface, but are prepared to tolerate shallow water.
Again, this might feel a bit too "transplanted from the surface" because they are surface plants, really, but that's the tedium of real life for you. This is your only way of getting fruits, I believe, as I'm not aware of any sea plants that produce fruits or fruit-like things. I may just be ignorant of them. You might make them up if you like, but there's no precedent for them, so it would depend on how tethered to reality you felt you needed to be.
Personally, I rather like the idea of merfolk cultivating surface plants in coastal areas, as a sort of mirror of surface folk cultivating seaweeds, but maybe its not your style.