Follow this guide to get to Bowser's door the quickest way possible.

Yoshi's Island 2:
Complete the level as usual. (Note: you may want to finish Yellow Switch Palace before moving on)

Yoshi's Island 3:
Complete the level as usual.

Yoshi's Island 4:
Complete the level as usual.

Iggy's Castle:
Complete the level as usual.

Donut Plains 1:
Here is the tricky bit. Grab a cape and fly. You'll eventually see a green pipe. Once you pass it stop flying and land on the yellow floor. Hold B and run into the key to pick it up. Keep holding B and bring it to the keyhole to move on.

Donut Secret 1:
Once you see a P block pick it up and swim to the keyhole. Press the P block near the keyhole and grab the new coins. The ? block holds the key. Open it and bring the key to the keyhole.

Donut Secret House:
Make sure you have a cape from Donut Plains 1 and go all the way to the right, through the door and all the way to the left. Pick up the P block, run and jump near the blocks. If you hold A the cape will carry you to a high area. Press the P block up here and a door will appear. Go through it and defeat the Big Boo (grab the exploding blocks by pressing B, hold up until the Big Boo appears and press B again to throw it.) to exit. Once out use the star road teleporter to reach Star World 1.

Star World 1:
Fall through the gap at the beginning and run to the far right. Spin jump (R) to reach both the key and the keyhole.

Star World 2:
When you reach the green pipe swim under it to reach the key and the keyhole.

Star World 3:
Use the Big Boo technique to knock Lakitu out of the sky at the beginning. Then hop into his cloud and go up to reach an area holding the key on the left and the keyhole on the right.

Star World 4:
Either grab a cape (Donut Plains 1) or a blue yoshi. (Star World 2. Feed the baby yoshi the falling star at the beginning) Then fly to the area on the bottom of the screen under 2 green pipes to find the key and the keyhole. Once out of the level use the star road teleporter.

There you have it. You're at Bowser's front door. Good luck!