After you chase Valentina and Dodo out of Nimbus Land, you can go to the hot springs to fall into Barrel Volcano.
Explore Barrel Volcano
From the entrance, go southwest to the next screen. Go to the southwest part of the area and carefully jump across the small platforms to reach the southeast exit. (If you fall into the lava, you don't take damage, but you are sent back to where you entered the room.)
Hit the two chests to get two Flowers. Beware that the Pyrospheres may come out of the lava and start chasing you.
After you get the Flowers, go northwest and jump back across the small platforms. Then go to the west to where there is a sign with an arrow on it. Jump up the stairs that the arrow is pointing at. At the top, go east and you will see another sign. Go northwest through the hole in the wall to reach the next room.
Make your way through this room with the Vomer (Dry Bones) and Magmus. In the next room, hit the chest to get a Flower, then go northeast and hit the chest to get coins. Go southeast from there, then northeast to the next room.
The chest in here contains a super star. Quickly go east, then at the fork go southwest. Then quickly jump up the stairs, and if you are fast enough, you can use the star to defeat the Corkpedite without having to fight it. Afterward, go west through the opening where the arrow is pointing.
Make your way through this straightforward room past the Vomers and Magmus. In the next room, to get the Frog Coin, stand northwest of the Frog Coin and jump onto it. You will hit the lava but you'll also get the Frog Coin on the way down. It can be difficult to find the right spot to jump from, so just keep trying. Afterward, go north to the next room.
Carefully go northeast and jump over the gaps. Pyrospheres will repeatedly fly out of the lava and attack. At the end, you may notice the Frog Coin above the archway. To get it, you have to go through that archway to the next room, then come back and fall into the lava on purpose. You will land on top of the archway, where you can get the Frog Coin. Afterward, go northeast to the next room.
You can optionally jump onto the floating platforms to get the Frog Coins in this room, but be aware that the platforms will fall after you stand on them for a short time. When you're done, go to the west side of the room and go southwest to the next room.
There are two Oerlikons in this room. Go southwest to the next room, then go northwest up the stairs, and hit the chest on the way to get a Flower. Hit the chest at the top to get a Frog Coin. Then save the game and go southeast. There is a Stumpet here. You can just run past it to the east and go through the archway that the arrow is pointing at, but if you do fight it, it is weak to ice and strong against fire.
In the next room, get past the Oerlikons and the Vomer, then when you reach the top of the room, go southwest to the next room. Go southeast and jump up the stairs, and you have to battle a Corkpedite. It's weak to ice and strong against fire. In the battle, it is made up of two parts. Aim for its body, then when the body dies, ignore the other part of it, because its other part will die soon after its body. Afterward, go through the archway that the Corkpedite was in front of.
Go northwest and you find another Stumpet. Go east from the Stumpet, then northeast, and you will see a sign with an arrow. Go northwest from the sign and follow the path to the next room.
Make your way to the top of the room. Hit the chest to get coins, then talk to the person in the archway. Hinopio will lead you into his shop. Stand at the left counter and talk to him to buy items, stand at the middle counter to use the inn, or stand at the right counter to buy armor. You can optionally go northeast to the east corner of the room to see some model F-Zero machines.
Use the southeast exit to leave Hinopio's shop. Go south and use the Save Block, then go northeast where the arrow is pointing. Carefully jump across the floating platforms, which will fall if you stay on them too long. Go northeast to the next room. There will be a cutscene, and you have to battle the Czar Dragon.