I've built PCBs with a standalone ATMega328p chip on them before for situations where I wanted all my electronics embedded onto a single board. It can be a great learning experience imo.
I would suggest first creating a schematic and implementing it on a breadboard with DIP packages as seen on the photo. You will need a crystal, loading caps (22 pF), as well as bypass caps (100nF) and pull-up resistors. Plus any other components your application requires.
After that typically you would burn a boot loader onto the chip so it can be programmed with the Arduino IDE.
Nick Gammon has provided an excellent tutorial on all of this. His article was my main resource when I made my PCBs: http://www.gammon.com.au/breadboard
Once you get everything working on a breadboard, you may want to look into laying out a PCB for a more permanent solution. Best of luck!