The following steps are given to search for new solutions to equations of sine-Gordon type. Step one, according to function transformation, the solving of sine-Gordon equation and sinh-Gordon equation is changed into the solving of two kinds of nonlinear ordinary differential equations. Step two, two kinds of nonlinear ordinary differential equations and quasi-Bcklund transformation of the first kind of elliptic equation are obtained. Finally, new infinite sequence solutions to equations of sine-Gordon type are constructed by applying Bcklund transformation and new solutions of the first kind of elliptic equation.