How can I refine my AE animation?

Just what I needed, thank you

So I ended up using both methods on the purple fish, applying the Wave Warp in the precomp, using the loop on the mesh in the main comp. Being a grouper I was going for a subtle effect, but what an amazing program.

The Gutterfish option is great, but here is another option, applying an eXpression. Either of the two options serves you.

Is the camera meant to be looking down from above or is it meant to be underwater with the fish.

The background looks like it's looking down from above but the fish is in profile.

If the camera is meant to be underwater with the fish then the overall path of the movement is unnatural.

There would not be so much movement in Y.   Fish tend to swim at pretty consistent depth.  Occasionally they ascend or descend drastically but not that often.

So those are the first two reasons it looks unnatural.  Third, is that any wiggling it's doing would be happening in Z, not Y..

In other words the wiggle would be toward and away from the camera, not up and down.

So if we are supposed to be underwater with the fish I would nix all the up and down movement entirely.

Make the fish 3D and have it orient along a path with a very subtle zdepth variation.

If the shot were from above (which is what it looks like) of course you would need a different fish drawing but the animation would much simpler because you could just add a subtle wave warp.

What you should do study some reference.  Watch some videos of fish swimming.

I'm not very professional with After Effects, but I think that a "oscillating" or "waving" path can be more natural. A fish normally do a "wave" path along the water. See if you can draw a "wave" INTO YOUR path (with pen tool). If you do not know what I'm refer, you can see this screenshot:

I'm believe this path can be more reallistic.
See this tutorial if you don't know how to apply this path to the fish (remember turn on the "auto orient" function"):

After Effects TEXT l OBJECT FOLLOW PATH and auto orient - YouTube

NOTE 1: You don't need to "Trim paths" or doing something similar, because we don't need to see the path of the fish.
NOTE 2: You can change the duration of the animation selecting all keyframes in the "Position" property and pressing "ALT" (Win) or "OPTION" (Mac) key and arranging the last keyframe to the necessary time.

Sorry if my english isn't very good. I hope you can understand.