Equipment or ingredients???
Equipment:
Bowls, spoons, electric (hand held?) whisk, measuring spoons (spoons and cup sizes), bun tins, cake and muffin cases, jug, scales, cutters, grater. pair of victoria sponge tins, loaf tin, large springform tin, square brownie tin, swiss roll tin?, baking sheets.
I put muffin cases in a bun tin to make muffins - fine if you don't require perfect looking, identical muffins.
Ingredients:
flour (plain, self raising, maybe strong or brown), sugar (brown, caster and icing), eggs, chocolate, chocolate drops, cocoa, margarine, butter, dried fruit (raisins, mixed fruit, mixed peal and cherries here, but currents, sultanas, apricots all substitutional, depending on your preferences!), , baking powder (maybe bicarbonate of soda and cream of tartar), vanilla essence (get the expensive stuff), ground almonds, selection of chopped nuts.
Thats what I normally have in stock, but I guess depending on your preferences for cooking you might want other things, and not use much of others.
Can't see why carrot cake would be that hard? MN has a fab recipe section, which is where I'd start if no books.