I gave your a crash course in moving Mario Yesterday but there is still some stuff you need to know before we can move forward.
Speaking of moving forward, You need to know about the P-Meter. Running fast builds up your P-Meter letting you do the biggest jump in any Mario game. For the record, the ‘P’ stands for ‘Power’ in this instance. This lets you dash over small gaps in the terrain and it also enables the Jet-Coaster Jump.
To survive trickier areas you’ll need to know how to do a Duck Slide. When a you see an area that only tiny-Mario can fit through, but you’re big this is how you get through. Run right at it and then duck while still holding right. You should slide forward just enough to get through. Failing that if you stand up in an area you don’t fit then you’ll just slip toward the right till you get free.
You can also squat while jumping. The upshot of that is your hittable area is halved so you can dodge all those Bullet-Bills near the end.