Nothing beats a Steadicam for capturing smooth video, but the commercially-available models can run to hundreds or thousands of dollars. If you're willing to do a little work, you can build your own Steadicam suitable for small camcorders and dSLRs for under $50.
The video above from Krotoflik walks you through the entire process. Most of the parts you need won't be lying around your house, but they aren't too hard to find online. A collection of washers at the base serves as a customisable counterweight to keep your rig balanced, while a bargain store flashlight is used as a clever free-rotating gymbal. The finished product looks surprisingly professional, and the sample footage linked at the end of the video is very impressive for something so inexpensive.
If you love this idea, but want something a little more compact for a GoPro camera, Krotoflik also offers a modified tutorial for the Krotocam Mini. Whichever model you choose, the video you capture should be leaps and bounds smoother than anything you could achieve by shooting handheld.