Play as Luigi
Beat the game then get all 120 stars by beating all purple comet missions then beat it again. Luigi runs faster and jumps higher than Mario, but stops slower than him.
Get the 121st Star
First, unlock Luigi, and then get all main 120 stars with him. Then talk to the main star, who'll ask if you want to go to Grand Finale galaxy. If you choose yes, you'll go there and get the chance to find the last star. Also, once you beat it as Luigi, you can play it with Mario.
Collect 9,999 Star Bits, the game's maximum amount, and all of the coconuts in the game will transform into watermelon!
Infinite 1-Up Mushrooms
This can be done in the Comet Observatory. Grab all of the 1-Up Mushrooms you can, which should be five (plus five if Princess Peach sent you a letter). Then, head into any dome, and then back out, without having played a level. All the 1-Up Mushrooms will be reset. Repeat as much as you would like