The fish that have a chance to give geodes, like Ice Pip and Lava Eel, are great, because they have other useful things, are rare, pretty visually and their roe sells for a nice little sum, when aged.
Sturgeon is a no brainer, because "caviar". Even a bundle requires it.
Rainbow Trout looks nice. It has an extremely low chance for Prismatic shard, but after ~80 real time hours, it finally broke and yielded me one. I say, well worth it, because even 1 of this extremely rare item, can make a diferrence (and for me, it did, hello "Crusader" on weapon with first enchant!).
Super Cucumber is another good looking sea creature. You can even use it in fish tanks! It gave me iridium a number of times and its amethyst, is loved by certain NPCs.
Blobfish and Spookfish are among my favorites. Pearl is very rare but i collect them, totem is handy and the Treasure Chest, is my favorite! I have two pirate hats, thanks to this little fella (spookfish)!
Lots of people recommend those. I myself expanded on this and added:
Octopus. Great for Omni Geodes, even though it yields them rarely.
Midnight Squid. We need a steady supply of ink, both for dyes and for cooking; as well as for Seafoam pudding, when he hunt difficult fish.
Blue Discus. Before i was able to get myself a banana sapling, this cute fish gave me some free fruit. It also yielded me a couple of Golden Walnuts, one of which contained the rare helm! What's there not to love?
Stingray. It has a very interesting item table, even though all of them are pretty rare. Only once it gave me a dragon tooth, but it was the single dragon tooth i had been away, from building the Obelisk. I take care of those who take care of me!
As a bonus, lava eel pond becomes red, in the color of lava. Willy loves octopuses, too.
I am considering to add a crab pond, because you get a reward and i like crustaceans. I am also considering eels, because lava eels rarely drop food. I riddled my field with fish ponds, as early as i could afford it! Invest in them fish right away, you will never regret it, i swear! Just remember to build lots of barrels in tandem, to age their roe!
from the Lava Eels and Iridium Ore 
and Amethyst 
could be useful for Omni Geodes. Blue Discus give Bananas and Golden Coconuts which are just fun! The Woodskip doesn't give particularly valuable things but delivers a decent variety of 
Wood and 
Tree-based things which is fun. The Stonefish does the same for 
Stone-based things. The Ghostfish does the same for 
