Most camcorders today record to a memory card or to a built in hard drive. Videos you record can be downloaded and saved onto you PC. This makes it easy to share videos with friends and family by uploading the video to sites like vimeo.com.
If you are worried about not having enough hard drive space, its much cheaper to buy an external USB hard drive than buy the mini DVD-Rs for the camcorder.
Canon, Panasonic and Sony make very good camcorders that are small, easy to use and feature packed. For about $300 you can get a good camcorder that records standard definition. For about $550 you can get one that records HD.
Now is a good time to buy a camcorder. New 2010 models just came out and many stores are discounting 2009 models. Sites like dealnews.com have one or two camcorders on sale almost everyday. Also check out SonyStyle's online outlet store.