I've been analyzing my golf swing for several years now. Since I don't have one of those fancy thousand dollar cameras, I realized that my "cheap" old iphone 4 would do the job. No, it doesn't give me 60 frames per second, but it does do 30 frames per second which I find is good enough to give me an idea of what I'm doing wrong.
Here's a sample video taken from my iphone 4:
There are other cameras that offer better frames per second. The cheapest are probably the Casio Exilim line of golf video cameras. For example, the Casio Exilim EX-FC100 costs about $200 on ebay. I'm not quite sure if they're worth it or not because although you do get up to 1,000 Frames Per Second, the quality really sucks. You need ample light to get a decent video at that rate.
One of the better options is to purchase a mini-DV video camera. Those start at around $500 or so, used. The Sony HDR-CX560V or the Canon ZR960 MiniDV are decent starter cameras for recording your golf swing. You'll also want a tripod to eliminate the shaky hands.
Another thing that I learned is that it's best to have your golf video camera about 6-10 feet behind you for a "down the line" video, and probably about 4-6 feet for a "face-on" video.
This is pretty much just the start of recording your own golf swing. The second, probably more important part is learning how to analyze it. I'm using the V1 Home software to compare my swings with a pro swing. It's a free software, but about $40 for the pro version, where you get some additional features. Note that the software only runs on a PC.
The great thing about the V1 software is that if your golf instructors uses the golf academy feature, you can actually send videos to your instructor and have them analyze your golf swing electronically. I actually did this with my golf instructor and he gave me some pretty good feedback...all electronically. Here's the video that he sent back to me:
Pretty cool shit, if you ask me.