Beside cooking times, how do I change a cake recipe to get good results in a microwave oven?
Here is a yellow cake recipe from a 1987 book of ours titled Step-By-Step Microwave Cook Book. I do think microwaves have changed a bit since then (wattages are higher), so you may need to adjust the timing.
Microwave Yellow Cake
1 cup all-purpose flour
1 teaspoon baking powder
1/8 teaspoon salt
1/3 cup butter
2/3 cup sugar
1/2 teaspoon vanilla
2/3 cup milk
Grease the bottom of an 8x1 ½-inch round baking dish and line with waxed paper. Set aside. Combine flour, baking powder, and salt. In a mixing bowl beat butter with an electric mixer for 30 seconds. Add sugar and vanilla. Beat till fluffy. Add egg. Beat till combined. Add flour mixture and milk alternately to beaten mixture, beating well. Spread evenly in the prepared dish.
Cook cake layer, uncovered, on 50% power for 10 minutes, giving dish a quarter turn every 3 minutes if turntable not present. If not done, cook on 100% power for 30 seconds to 2 minutes or till the surface is nearly dry.
Cool cake on a wire rack 5 minutes. To loosen, run a narrow metal spatula around dish edges and slightly under waxed paper. Invert onto rack. Remove waxed paper. Cool and frost as desired.