With the ocean going life its fairly simple, they swim around the continental edges.
Most rivers eventually end at the ocean, so fish that evolved a tolerance to freshwater could swim up these rivers, and then into any tributaries and lakes from that.
Seasonal flooding can temporarily connect rivers and lakes that didn’t used to be connected, and additionally the ever changing paths of rivers as they carve out the landscape can slowly introduce species to new areas. At one point in the past all the continents were connected to each other, so a lot of species spread out across the super continent then and then evolved more on their own once the continents split. This is why for example you can find species of Tetras or Cichlids in both south american and african waters.
Aquatic plants are a bit different of a story. Most aquatic plants started out as terrestrial plants that have then colonized freshwater, and then saltwater to a degree. Most seaweeds you find in the ocean like kelp aren’t considered true plants, but rather types of colonial algae.
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