After you talk to the Chancellor, you meet Mallow and decide to help him catch Croco.
There are Goombas and Spikeys in this area that will rush toward you if you get close. When you battle them, you may encounter Frogogs as well. Go southeast in this area to find a save block. Save there, then go southeast to the next area.
There will be some brief dialogue from Croco. Afterward, jump onto the spinning flower, and use it to jump northeast. Then carefully jump onto the floating platforms to get the coins. The floating platforms will rotate whenever you jump, allowing you to go all the way across. You can optionally go down below to battle the K-9s. Then you can jump back up onto the floating platform, and jump east onto the hill. Before you go to the next area, jump onto the spinning flower to the east of the hill, and use it to jump east. You will hit a hidden treasure chest and get a Croaka Cola/KeroKeroCola. Save this item for later. You can use it in battle to heal up all of the HP and FP of all party members, but it doesn't work on fallen party members.
Then go east to the next area. After more dialogue from Croco, make your way north. When the path splits, go northeast to find a chest that is guarded by a K-9. If you defeat the K-9, you can open the chest to get healed up by a healing mushroom.
Then go back to where the path split, and go northwest to reach the next area.
There will be some more dialogue from Croco. Jump onto the moving platform, and use it to reach the treasure chest. Open the chest to get a Starman. This makes you temporarily invincible, and you will defeat enemies as soon as you make contact with them, and a bunch of enemies will appear while the Starman is in effect, so run around and defeat all of the enemies before the Starman effect wears off.
Afterward, you can use the floating platforms to reach the treasure chest to the east, which contains a Flower.
Then go east to the next area.
After Croco runs away, go east and open the chest to get healed up by a Mushroom. This area has Koopas/Paratroopas flying around, as well as some hidden Goombas that will jump out at you. Try to sneak up behind Croco. After you catch him three times, you will fight Croco.