Goomba Mountain is the first course in Super Mario 64: New Castle. It is accessed by going left from the entrance of Peach's Castle and entering the door with a Goomba on it. You do not need any stars to access it.
Star 1: Goomboss's Anger Surge
To complete this star, you have to take the dirt road to the mountain, bypassing the various stone paths on the way. Then you have to climb up the mountain. When you get to the top, enter the small door leading inside the wooden arena to meet Goomboss. Defeating Goomboss is simple - you have to stomp on him several times. Each time he gets angrier and faster. Upon defeating him, he will hand over the Power Star he was guarding.
Star 2: Ultimate Racing Tournament
Near the starting point for this star is a racing tournament. Enter the tournament to find that you are racing Koopa the Quick, Il Piantissimo, a Toad in a go-kart and Goomboss. The race goes around the mountain and then up it to the goal, a huge circle where the arena used to be. Koopa the Quick and Goomboss are the slowest, however Goomboss gradually speeds up the further you are ahead of him. Il Piantissimo is moderate speed and the Toad goes the fastest. To get the Power Star offered up as a prize, Mario has to come in first place.
Star 3: Goomba Plane
To the left of Goomba Mountain is a cannon. Get in the cannon and fire at the Goomba Plane which is dropping Bob-ombs. Upon hitting the plane, it crashes down. The Goomba offers up a Power Star.
Star 4: Red Coins of the Mountain
For this Star. Mario has to collect eight red coins. Half of them are to the right and left of the mountain, and half of them are on the mountain. The Star will appear at the start of the course after you collect them all.
Star 5: Ultimate Sunlight
The sun is a lot brighter than usual! This causes all the enemies to be tougher, but slower. Mario must have completed the Secret Course with the Wing Cap to get this star. If he has the Wing Cap, Mario must go to the top of the mountain and use it to fly through the funnels of light. If he can stay in the funnels of light without missing any, he will be rocketed up to the sun. It turns out the reason the sun was brighter was because a Power Star was by it, providing lots of light. As soon as Mario reaches the sun, he gets the star.
Star 6: Play with Chain Chomp!
To the right of the mountain is a Chain Chomp. Normally the Chain Chomp would attack you, but on this star he wants to play. He requests you to bring him food and toys from across the course. Eventually, he will thank you and jump off into the distance. You can now ground pound his special pole, which contains a Power Star.