I am assigned by my teacher to do a super mario project on scratch, I don't know how to make my super mario sprite run while changing costumes and jump while changing costumes. But also when I jump up the step, I don't know how to keep the sprite stay there because if I do this:

The sprite won't go up the stairs, so I really need some help please.

There are platformer tutorials for this. I will give you the links to some of them, but here's something to think about: you need a script to stop when it is touching a certain sprite or color, so you could use a variable to change the position of the character, and manipulate that variable instead of using change x/y blocks. Try to understand the code in these tutorials instead of just backpacking the code, so your teacher doesn't get angry at you.

griffpatch_tutor's tutorial (complicated but amazing results).
djpro's tutorial (simple and effective).

Basic Platformer Wiki Tutorial.
Advanced Platformer Wiki Tutorial.

Thx

How do you make a background move like Super Mario?

youre jumping up then jumping down right away. that won't work. You might have forgotten to put the forever loop in. Maybe try: