The Angler Hut is located in Sansee, and the Vivarium is located in the Wash.

Once acquiring the fishing rod, press T to start fishing. Cast your line with E and press E to reel it in quickly when there is a bite. Then, follow the fish icon with the A and D keys until the small bar turns completely blue - you've caught yourself a fish! This will also net you the Quite The Catch achievement when done successfully for the first time.

On controller, press right on the d-pad to start fishing, cast the line and reel in with X, and follow the fish icon with the left stick.

You can also turn on Easy Fishing in the controls menu, which will prevent the bar from getting red and the fish from getting away.

These hints come from Researcher Casii in the Vivarium.
Bellow Guppy: Common in the sands of Midden. Hard to miss.
Bloated Eye Clam: A parasitic tree killer.
Dopey Dart: Extremely common in the Sansees.
Flutter Thrust: In the heart of Midden, below pillars of rock.
Ghost Soul: Seek the souls of old bones.
Hummer: These fish seek green areas with trees and foliage.
Light Koi: A rare find, these will often feed on shrubbery in the Badlands.
Little Kicker: A common find in pink colored dunes.
Pebble Eye: A common fish that seeks shade in the Eastern Dunes.
Pill Fish: Persistence will pay off when fishing in areas full of cacti.
Sand Ray: In Redsee, these fish seek the deeper parts of the sands.
Scabby Fish: A common find in the sand paths between towering cliffs.
Shelled Cup: Seems this creature feeds on salt.
Storm Fish: A tricky catch that rides in the crests of crimson sands.
Sucker Fish: Seek the shade of large rock arches in Redsee.
Teacup Angler: Rumor is these creatures feed on gaseous substances.
Trifle Jelly: Likes pale mounds of chalk.

Bellow Guppy: found everywhere in Sansee.
Bloated Eye Clam: found near the trees of the Petrified Forest in the Wash.
Dopey Dart: found eveywhere in Sansee.
Flutter Thrust: found in the middle of the Badlands, next to the tall pillars.
Ghost Soul: found in the field of bones in the Wash.
Hummer: found in the forest in Redsee.
Light Koi: found by any shrubbery in the Badlands.
Little Kicker: found everywhere in the Wash.
Pebble Eye: found in shadow or during the night in Sansee.
Pill Fish: found next to cacti, there is a field of them where Sansee meets the Waste.
Sand Ray: found in the middle of Redsee, between Eccria and the Cartographer's Balloon.
Scabby Fish: found anywhere in the Badlands.
Shelled Cup: found anywhere in the Waste.
Storm Fish: found anywhere in Redsee.
Sucker Fish: found at the rock arches left of Balfron Connection in Redsee.
Teacup Angler: found inside a gas vent close to Glow Worm Cave in Hakoa.
Trifle Jelly: found close to any of the white dunes in the Wash.

For the first leg of the quest, you will need to catch three fish in Sansee: a Pebble Eye, a Bellow Guppy and a Pill Fish.
Turning in these fish to Yoshi will net you the first Angler's badge!

For the next one, you will need to find a Little Kicker, Trifle Jelly, and a Bloated Eye Clam from the Wash.
Turning in these fish to Yoshi will net you the second Angler's badge!

For the last assignment, you will need a Storm Fish, a Sand Ray and a Hummer. These are found in Redsee.
Turn them in to Yoshi for the last time, and congrats! You can go and make your Angler's mask! Doing so will net you the achievement The Angler Mask.

You will have to collect every single butterfly and fish for Researcher Casii in the Vivarium. She will provide hints for the fish (for a price), but I've also listed them above. She will also remind you which fish or butterflies you have not yet turned in. After you give her a number of fish, she will give you another fishing rod which will make fishing easier. The butterflies you have to collect for her and their location are as follows:

Chalk Butterfly: Found in the Wash.
Crystal Butterfly: Found in Hakoa by the crystals.
Nimoor Butterfly: Found in the forest in the Wash at night. Grab the blue glowing things and throw them.
Oasis Dragonfly: Found in large open bodies of water.
Salt Butterfly: Found by flowers in the Waste.
Shaded Leaf Butterfly: Found in the palm forest in the western dunes.
Stepwell Dragonfly: Found in the Thieving Magpie Well.
Sunshine Butterfly: Only found during the day.

After you turn in a specimen, it will show up in the Vivarium. When you turn all of them in, you will get the Vivacious Vivarium achievement.

Once you fill the Vivarium with all the butterflies, Merchant Sola will appear in it. They will sell 3 dragonfly bike parts for 150 each, a butterfly top for 400 and butterfly trousers for 250.

I have not found the new dyes yet.

Thank you for reading, and I hope I was able to help! If you enjoyed the guide, rating it up and favoriting it goes a long way.