Richardson, 33, was reunited with his former mentor Paul Cook this week when he left the Spireites' League Two rivals Accrington to make the switch to north east Derbyshire.
Cook himself made the same move in October when he succeeded John Sheridan as Chesterfield boss.
Richardson has been impressed with the Spireites' set-up and believes the club should be playing at a higher level.
"Chesterfield is a fantastic club that should be knocking on the door of the Championship, and hopefully we can go about doing that swiftly," said the former Accrington manager.
"The infrastructure already in place here is impressive and the ambition of the chairman and the chief executive is that they want to be up there.
"I'm young myself and enthusiastic about progressing my own career. I want to be in that same place."