After you defeat Belome, you are washed away by a flood and have to navigate the Midas River.
Navigate the Waterfall
After you defeat Belome, there will be a cutscene. Afterward, there will be a minigame where you will go down a waterfall, then a river, where you can collect coins. At the end, you can exchange the coins for Frog Coins, and any coins you collected so far will be saved if you don't have enough to exchange. You can say Yes to ask Toad for an explanation: press left or right to move side to side on the waterfall, and tap the jump button to briefly swim upward.
You will be able to play this minigame again repeatedly, so don't worry about doing it perfectly. If you don't feel like playing it, you can just not touch any buttons, and you will reach the end of the course either way.
At the start, stay to the right so you can get the frog coin. You can optionally fall into the cave, but there is nothing in it but some scenery to look at, and you will end up beneath the cave entrance after a brief detour.
After that, stay to the left to get more coins and another frog coin. If you go into the cave here, you get a Flower Tab but miss out on the remaining coins in the waterfall. If you didn't go into the Flower Tab cave, skip to the next paragraph. If you went into the Flower Tab cave, there will be another split in the waterfall path, and if you go to the right at the split and go into the cave, you get another Flower Tab.
If you didn't go into the Flower Tab cave, you can get a frog coin, then fall into the cave beneath it to get another frog coin. You will end up beneath the cave entrance afterward. Then if you stay to the right, you can get a final frog coin.
Jump on Barrels
After the waterfall, you can get an explanation from Toad about the barrel jumping event: Jump to collect the coins above you. There are two sides to the river. If a barrel is coming toward you and you don't do anything, you will be moved to the other side. If you jump instead, you will stay on the current side. Be sure to jump to avoid getting hit by fish, or you will lose coins.
Get a Gift
At the end, if you got enough coins, you can exchange them for frog coins. If not, your collected coins will be saved for later, so you can come back and try to get more coins.
The first time you complete the course, you also get a NokNok Shell, which is a weapon that Mario can equip. You can talk to the person or jump on the spring to try the minigame again. It costs 30 coins for each attempt.
Save at the save block, then go southeast to Tadpole Pond.