After you defeat the Hammer Bros. in Mushroom Way, you arrive in Mushroom Kingdom.
Visit the Shop
In the eastern part of the town, you will find a shop where you can buy some helpful items, like a shirt that increases your defense, and some helpful accessories, and items that you can use both in and out of battle. Don't sell the Hammer that you got from the Hammer Bros. It is a unique item and you can't get it again if you sell it or throw it away. (You can't sell it in the Nintendo Switch version.)
Talk to the person in the basement of the item shop and follow their instructions to find a hidden treasure box that contains a Flower, which increases your maximum Flower Points by 1. In the Nintendo Switch version, this person will also give you a Signal Ring, which gives an audible and visual indication if there is a hidden treasure nearby. There is another hidden treasure box in this same room. Stand on the person's head and wait for them to go to the eastern corner, and jump there. This treasure box also contains a Flower.
Visit the Inn
You don't need to rest in the inn right now, because you will be healed up for free soon, but you can save the game there. The inn is northwest of the shop.
Visit the Castle
Go to the north part of town to find the castle, but don't go in yet.
Missable Treasure Chest Warning!
In the Super NES version only, there is a hidden treasure chest in the castle that you can permanently miss if you don't get it now. See below for instructions on finding it. In the Nintendo Switch version, even if you miss the first opportunity to reach this chest, you can still get it later.
Enter the castle. To avoid missing the aforementioned treasure chest, when Toad starts walking away, jump onto Toad's head. Before Toad goes through the door, jump northeast onto the top of the stone wall. In the Super NES version, if you don't make it, make sure not to go into the throne room. You can leave the castle and go back in to make Toad come back so you can try again.
After you jump onto the ledge, in the Super NES version, go east so you are standing above the right side of the door (but not all the way in the east corner), and jump up to find the hidden treasure chest. In the Nintendo Switch version, go to the middle of the ledge above the door. The chest is near the northeast wall. Hit it to get a Frog Coin. You can't use Frog Coins yet, but they will be helpful later.
Afterward, you can explore the castle. To the northwest, you find a corridor that leads to Toadstool's room. In Toadstool's room, if you check between the chair and the fireplace, the chambermaid gives you a Mushroom. The southeast room leads to the guest chamber and vault, but you can't do anything there yet.
Go to the northeast room (where Toad went) when you are ready. Talk to the Chancellor and there will be a cutscene. When you try to leave, there will be more dialogue, and in the Super NES version you get a Map. You can press the menu button to check it. In the Nintendo Switch version, the Chancellor asks if you would like an explanation of how to equip items.
Afterward, go to the southeast room of the castle, then go southwest and talk to the guard to enter the Vault. You can open the chests to get ten coins on the left, a Flower in the middle, and a Mushroom on the right. You can talk to the vault guard to hear an explanation of Flower Points if you wish.
Meet Someone New
When you go outside and walk away from the castle, there will be a cutscene. Talk to the crying person. When a bystander asks you to show how you will defeat Bowser, press the jump button. Agree to help Mallow.
Go to the shop and talk to the shopkeeper, who will give you a free Pick Me Up, which revives a fallen ally in battle. Afterward, talk to the shopkeeper and buy the Pants for Mallow if you didn't get them earlier.
In the Nintendo Switch version, if you missed the hidden chest in the castle, you can now go back in and stand on the Toad near the throne room door to jump up on the ledge, stand in the middle of it near the northeast wall, and jump to find the hidden chest.
Talk to the East Guard
Go to the east part of town and talk to the guard who is jumping up and down. There will be a cutscene. Afterward, in the Nintendo Switch version, the guard will give you a map, which you can use to instantly go to places you have been before. Save in the inn if you wish, then go east to Bandit's Way.