After the introduction cinema, you'll appear inside Bowser's Keep as Mario. In the first room, simply go through the door to proceed.
The next room features a number of Terrapins walking around. They'll be no trouble to you, but you might just as well avoid the battles by evading or jumping over them. At the end you'll be challenged to a fight against four Terrapins. This fight is rather easy, just attack the Terrapins using and they should be dead quickly.
You'll be automatically healed after this battle. Go through the next door and cross the bridge over the lava pit. Watch as it crumbles just behind your feet, then proceed into the next room where you'll see the Princess hanging down from the ceiling and Bowser on one of the chandeliers. You'll then enter a fight with him.
You can attack Bowser but he won't take any damage. Instead, your target is the Kinklink holding the chandelier Bowser's standing on. It's best to perform a jump attack - to do so, press to perform a Special attack, select the skill "Jump" using the same button, use your Control Pad to select the Kinklink and press again to attack. If you can do timed attacks the Kinklink should go down in two hits. If not, it might take a while longer, but overall it should be easy.
Once the Kinklink is dead another long cinema will play. Bowser will crash down while Mario jumps up to save the princess. But then, the castle will suddenly start to shake violently as a giant sword crushes down into the castle and sends everyone flying.
Mario will crush down into his own house. After a short talk with Toad, press to jump out. As a general rule, if you find yourself stuck in some situation where you can't move or hit , it's usually because you have to jump to progress the plot for whatever reason.
You can check out the lamp to rest, which fills up your HP and FP fully (more on that later). But now, leave your house and talk to Toad using . He'll tell you to go back to the castle, but before you do this, jump on the floating platform with the multicolored S to save your game (it's required anyway). Once you've done that, leave the area to the south, and on the world map, go back to Bowser's Keep. This time, the giant sword that recently crushed into the castle destroys the bridge that leads to the castle, leaving Mario stranded.
After this scene you'll automatically reappear on the world map, so go back to Mario's Pad. You'll find Toad inside your house, so go inside and talk to him. Mario will now retell the story, after which Toad suggests talking to the Chancellor. Try to exit the area again and Toad will bump into you, but give you a Mushroom in exchange. A mushroom restores 30 HP and is an invaluable item in early game. Toad will then ask you whether you know how to use items. If you select No a tutorial will commence which will have you eating the Mushroom you just got, so select Yes. Toad will then ask you whether you know what Timed Hits are. Feel free to take the tutorial if you don't have any idea, it doesn't hurt. You'll then get another three Mushrooms, after which Toad will rush off. Follow him into Mushroom Way.