In this paper, we investigate the unified method and the modified Kudryashov method for obtaining exact traveling wave solutions of conformable fractional partial differential equations. In addition, a connection is given between the two methods. Then, using these methods, we obtain new exact solutions for the space-time fractional (2 + 1)-dimensional Calogero-Bogoyavlensky-Schiff equation. Fractional derivatives are described in conformable sense. Various solutions have been obtained, including one-soliton, kink, anti-kink, periodic wave solutions, and multiple-soliton solutions. We also provide a graphical representation of some interesting exact solutions to the equation and discuss the behavior of these solutions. The considered methods can be effectively applied to a wide range of nonlinear fractional partial differential equations. [ABSTRACT FROM AUTHOR]