Water is 1000x denser than air, and much more viscous. So saddling fish only works if you are going to be riding on the surface. That's on top of the fish having a shape that allows for saddling.
If you are going to be underwater, you'd better be horizontal and grab the animal by some means. If they have shark skin, you might be able to hold on to them (you'll need a thick skin or you may bleed).
If you wish for something more natural than a saddle and hardier than plants, remoras will do:
Or if the fish is male, you can hold on to them by the linus. Sharks and rays have two of these. This may also be used to cause the fish to release some nutritive, protein-rich secretion.
Aerial fish may be surfed on. There is plenty of literature on this:
And here is a real life example, as seen in Sharknado 2: The Second One:
In the comments, user3819867 called to my attention that just you might also like to ride inside a whale's innards. While whales are technically not fish, they are the best marine fauna for this mode of transportation.
Real life whales have their throats too narrow to swallow an adult. A workaround would be having them swallow a human infant, which would them grow up inside the beast.
But since we've already delved into DC physics here, you may wish to get inside marine fauna as an adult and ride it from there. I searched for images of people getting inside sharks and whales but everything I got was gory, so have an artistic rendition instead:
Then again, that's not much controllable. You might want to do like they do it in Windforge, a game which is all about hunting whales for sport and turning their carcasses into flying contraptions:
Don't buy that game though. The developers abandoned it halfway through development.
Finally, if you really.wish to invest in fish husbandry, you can start with barreleyes, which have transparent heads:
And turn them into submarimons: