I was inspired by the channel Ben Eater, and his videos on making various components of an 8-bit breadboard computer. (https://youtu.be/wvJc9CZcvBc) I watched his video where he made a binary 4-bit adder using nothing but AND, OR, and XOR gates. I took his logic diagram, bought some components, and made it on breadboard.
I learned a lot from my mistakes while making it, such as the fact that you cannot simply turn off a switch, stopping the current flow for logic 0; you must have a pull-down resistor!
Above is a picture of the finished project. The DIP switch on the left makes up the two inputs. Here, the input is 0010 + 1100. The output, as you can see, is 01110, as it should be!