A few years ago I switched to wet shaving and in doing so I saved money, got rid of my razor burn and now get a much closer shave.
Wet shaving involves using a badger brush and good ol' shaving soap. You soak the badger brush with water and coat it with shaving soap. You then spend a few minutes working up a nice thick lather. I form the lather directly on my face, but many folks prefer to use a separate bowl to do so.
Don't be fooled by the photo below, I actually use handmade natural soap from Classic Shaving. I just happen to put it in a vintage Old Spice mug. The shaving soap lasts for months and is much cheaper and more effective than the store bought glop in a can.
For the actual shaving I use a vintage Gillette safety razor. It has a nice large handle and feels well balanced in your hand. It's called the Fat Boy and is especially popular with vintage shave enthusiasts.
The blades I use are made by a Japanese company called Feather. They're pretty much the sharpest razor blades you can buy and they work wonders. You can get a pack of ten for about six bucks from Classic Shaving. That equals big savings if you consider that Gillette five pack of Mach 3 blades will run you around $15.
Regardless of price, the sharper blades combined with the badger brush and soap have eliminated the razor burn I used to get. Plus, wet shaving is fun!