This is my Go-To cake recipe. I've used it at least 30 times, with dozens of different flavours, and passed it on to friends, who've also made it over and over. It's super easy, and INCREDIBLY versatile! Oh and it works well for cupcakes, too.
Quick Plain Cake
3/4c sugar
1 1/2c flour (plain or all-purpose)
1/4t salt
2t baking powder
1/4c melted butter (I often just use a mild oil instead)
1 egg, beaten
1c milk
1t flavouring (can be whatever you want - vanillas essence, citrus zest, cinnamon, melted chocolate, booze, whatever. If I'm doing zest then I'll do the whole fruit then add some of the juice to make the 1t of wet)
Put all the dry stuff in a big bowl and stir so the baking powder is evenly distributed. Drop the beaten egg in to the melted butter (or oil), then add the milk and flavourings. Pour the wet ingredients into the dry ones and mix together with a wooden spoon. Pour into grased and floured pans and bake at 350F (that's about 180C) until the cake starts to come away from the sides of the pan and a skewer comes out clean.
Cool on a wire rack, and be sure to COOL COMPLETELY before attempting to icing.
Oh what the heck, since I'm typing this all out already, here's my stupidly easy buttercream icing, too:
Melt some butter in a big bowl in the microwave. Add in a few T of milk or booze or maply syrup or some other liquid. Mix together with a spoon. Then just keep adding powdered sugar, stirring the whole time, until you get the right icing consistency. If it gets too thick, add a splash of milk. Too runny? Keep adding icing sugar.
Et voila, a birthday cake.