How do I get past the giant Spider?

There are two ways. First off, his movements: He'll raise his left leg twice, then move to the right a little, move to the left a little, then bend his legs and get his head closer to the ground. That's all you need to know on that part.

The first way is getting between his legs when he raises his left one either the first time or the second time, move with him when he sidles to the right, then left, then, when he bends his legs, jump up and attack it's face. (This is not recommended on the asteroid belt stage in the Starfruit Kingdom because of the ice. The chances of hitting it's legs are very high on that stage.)

There's a second part to this after you get to it's head, but we'll get to that later.

The second way, which is a little harder, but faster, is get between it's legs THE FIRST TIME ONLY it raises it's left leg. If you try it during the second time it raises it's leg, you won't get to the face in time. Then, while it's lowering it's leg, do a backwards somersault (Z+A), and the extra height from that special jump gets you to the spider's face early.

Now for the second part, after you attach yourself to it's head. There are three eyes that'll blink in different intervals. You can only stun the spider while it's middle eye is open. Hit that, then you can vent out your frustrations on that spider with some hardcore pummeling.

Hope this helps.