Upgrading tires on my road bike I'm very new to the complex world of bicycles, so apologies ahead of time...

I have a 2012 Giant Defy 1 (https://www.giant-bicycles.com/us/defy-1-2012) and am looking to upgrade the wheels to something more commuter friendly. The tires on my road bike are stock 700 x 23 and the rims are stock Giant PR-2.

My questions are:  


What's the best size tires to use? I read that 32's are too wide,
but my worry is that 26's might be too narrow.
Will I need to get new rims to accommodate the larger tire or will
the stock PR-2 discs, which currently use 23's, work?
Well, 26 will feel a whole lot better than your current 23. You can get away with lower air pressure because of the larger volume supporting your weight. Lower pressure means smoother ride. Of course two tires of the same size can feel very different because of the quality of the casing and rubber. For commuting, I would say get as big as you can fit, which is hard to know without seeing the bike. Many modern road bikes can fit up to 28mm, but you'll have to look and see. There are three potential spots to look at: fork, rear brake bridge, and rear chainstays. If you can measure the gap between your current tire and each of those three spots, you can get a very rough idea of how much bigger you can go.

Your rims should be fine with anything at least up to 32mm if not more. They can accommodate a fairly wide range of tire widths. As the tire gets wider, it will tend to make more of a lightbulb-type shape which can be less stable, but that usually isn't bad until you get extreme in the differences.