One easy option for temporary rainbow hair is chalk. "Chalking" has become super popular because of how easy it is to wash out of your hair, and you can use a lot of different colors at once. I like using chalk because it shows up on even dark hair, but it's sort of messy and always made my hair look super dry. If you want to try chalking at home, make sure you're purchasing soft chalk pastels, not oil pastels, at the craft store. Free People even makes a cute set of chalks for your hair. My favorite tutorial to follow can be found here.
Another version of "chalking" is color rub, which looks like eyeshadow, and you apply it much the same way you do chalk. My preferred option for color rub is Kevin Murphy's Color Bug
Another option is semi-permanent dye. Manic Panic is my favorite (probably since it's the only one I've used...). It is more difficult because, unless your hair is very blonde, you have to bleach it first. But Manic Panic offers a wide array of colors and it doesn't feel so damaging to your hair. I used this tutorial 2 years ago and it's still the best one I've found.