Super Mario RPG: Legend of the Seven Stars/Bowser's Keep

After the cinematic, you'll have control of Mario and be able to hop around as you're thrust into the thick of things. Bowser's troops wander around at random, policing the doors and bumping into the statues on the wall (the enemies won't follow or trick you, but keep your distance)

The castle halls are lined with TerraPins, little 10 HP monsters you should be able to kill with no trouble. If you used timed defenses you can make it through this whole section without taking any damage, otherwise just make your way through the doors, fighting the enemies you have to and avoiding the ones you don't.

You'll come face to face with Bowser atop two Chandeliers, with the princess looking on, tied from the ceiling. You can heckle Bowser if you want but it won't do any good. Just jump or attack the Kinklink to deplete its 65 HP and send Bowser's Chandelier crashing down. Mucho plot related stuff will happen, and you'll end up back at Mario's pad.

Mario's Pad
After meeting Toad you can hit to jump out of bed when you're ready. 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.

While at Mario's Pad, you can rest in the bed all you want by turning off the light, save your game, and eventually head back to Bowser's Castle to see what's up. Enjoy your first view of the minimap and select Bowser's Castle to find a giant sword announcing the arrival of the Smithy gang. The bridge to Bowser's Castle will be knocked out, and you should head back to Mario's Pad to meet up with Toad.

Toad will send you off to the Mushroom Kingdom after giving you a few mushrooms and a tutorial on items and timed hits if you want them. The item tutorial will make you lose one of your mushrooms, however you might want to check out the tutorials if this is your first game. After that, head on into the next chapter.