Depending on the context, "Methodist" could be a noun, a part of a proper noun, or an adjective.
Methodist should always be capitalized.
Noun:
Susie is a Methodist.
Part of a Proper Noun:
I am a member of The United Methodist Church.
I was born at Methodist Hospital.
Adjective:
That is a Methodist church.
Prevenient grace is a distinctive belief of Methodist theology.
John Wesley is the founder of the Methodist movement.
I was born at a Methodist hospital.