Rinse the rice until the water runs mostly clear. Cook it with the water according to the rice cooker or saucepan instructions.
Warm the vinegar, sugar, and salt just until dissolved. Fold the mixture into the hot rice and cool it to room temperature.
Roll the Sushi:
Place a sheet of nori shiny-side down on a bamboo mat. Spread a thin, even layer of rice over the nori, leaving a 1-inch strip clear at the far edge.
Arrange fish, cucumber, and avocado in a narrow line across the lower third. Lift the mat and roll firmly, pressing the filling inward as the nori closes.
Moisten the clear edge to seal the roll, then slice it into 6 to 8 pieces with a damp sharp knife.
Serve:
Arrange the sushi rolls on a platter and serve with soy sauce, pickled ginger, and wasabi.
Notes
Use sushi-grade fish from a trusted source.
Cool the seasoned rice before rolling so the nori stays crisp.
Keep fillings narrow for rolls that slice cleanly.