In a nutshell: the correct proportions, the right footwear and a protective top layer. But this doesn't mean slickers + Wellies are the only options.
Starting with the proportions of your outfit, you want skinny on the bottom and bigger on the top (don’t want any pants dragging into puddles).
Next, choose your footwear: leather boots or something with a heel, wedge or lug sole (rubber rain boots do keep you dry, but they get awfully hot once inside). You want to be literally elevated off the ground. A relatively inexpensive dark leather boot often does the trick--you don't mind if it gets a little wet or marked up. If you prefer to invest a bit more money, choose a brand with weather-proofed styles like Sorel or Aquatalia.
The following step is to layer up on your coats. If your raincoat is thin, but waterproof, wear it over top of a light weight puffer vest or jean jacket or wool coat to add warmth. Of course, if it's one of those warmer, wet days (also known as 'great hair days' LOL), a thin coat is all you need. It doesn't even have to be waterproof because you are going to have a great umbrella :)
And, finally don't forget the power of color on a gray and rainy day. A bright rain hat or umbrella is your finishing touch. Time to start singing in the rain...just singing in the rain...
For easy purchasing or just for a closer inspection, click here for links to all of the above images.