This is Michigan State University. Our colors are green and white. We are the Spartans.
Our sports rivals are University of Michigan (“U of M”). They are in Ann Arbor; their colors are blue and gold; and they are the Wolverines. There is a lot of mostly-good-natured rivalry between MSU and UofM, but don’t call MSU just “Michigan.”
The State of Michigan is refers to the state in general or the state government.
People from Michigan call themselves Michiganders (much more common) or sometimes Michiganians. And they will use their left hand to point out directions in Michigan on their right hand. (The lower peninsula of Michigan looks like a hand).
Lansing is the capital of the state of Michigan. East Lansing is a separate city next to Lansing.