In this paper, we focus on geometry of evolutoids and pedaloids for timelike curves and spacelike curves on timelike surfaces. First, by studying an example of a pseudo-circle, we get some properties of timelike curve on timelike surface. Then, we introduce a moving frame along curves and describe the corresponding evolutoids of curves as ruled surfaces. After that, we investigate relationships between pedal surfaces of evolutoids and pedaloids of curves. Moreover, we investigate the singularities of the evolutoids and the pedaloids.