I want a button that, when pressed five times, outputs a signal. What redstone device should I use?
What you're describing is called a unidirectional ring counter. It's a circuit with (in your case) five outputs, with only one ever on at the same time. Each time the circuit receives a pulse, the output that is on moves over by one. When it gets to the end, it starts over at the beginning again. You'll only be using the fifth output, but you need the rest anyway just so that it can count your button presses.
The design in this video tutorial is fairly compact, easy to understand, and easily extensible (up to the limit that pistons can push). It uses a pair of pistons on either end to cycle blocks between them; most of the blocks are glass, with two being a block type that will pass a signal from a redstone torch below to a repeater beside them. Some people don't like using pistons since they're noisy, though, so that's a possible drawback of this design.
The pure-redstone alternative is chaining five T Flip-Flops, and connecting the output of the last one to both your final output and the input of the first T Flip-Flop (so that it will start over once you get past 5 presses). Any of the designs at that page will work, so you can choose whichever will fit the space you have to work with.
Another, maybe simpler, way to accomplish this is this simple piston line.
It is just the same loop repeated
Things to note about this method: