How can I build a board that I can save a program on to, to install into my project so that I can just plug and play. Therefore I wouldn't need an arduino board everytime I operate my prototype.
Arduino is not good for your need? It is a prototype board.
Buy an 'Arduino Pro Mini'
This will be sufficient for many projects.
Can use an UNO to program these but you should get one FTDI as a dedicated programmer.
I assume you wanted to build a board with a standalone Arduino, that means you dont need an Arduino board in your board.
I used this schematic to create a system which can program using a USB FTDI cable.
See the board I developed recently.
I'm with LarryD on this one; the 328P mini from China is your best bang for the buck.