I don't have a camcorder, so I figure I might as well get a go pro, so I can take it with me for biking. As far as I know, there are three places to best put it: helmet, handlebar, chest.

handlebar = rattling screen from the aluminum handlebar bouncing up and down
chest = different views from the hood vs aero positions
helmet = less helmet ventilation, heavier helmet

I'm leaning towards helmet, toward the front of it, because it offers more of a 1st person perspective. But I'm wondering what others think.

I have all three mounts and by far prefer the chest mount. It results in far steadier video that either the handlebar or helmet mount, and you hardly notice it's there.
The handlebar mount is kind of scary as it does not look terribly secure, and more drive train noise is picked up.

I find videos made using a helmet mount to be nauseating.

I usually place mine on the handlebars. But what I think looks nicer but is more difficult to achieve is mounting the camera to the frame so it is not affected by handlebar movement. The little side to side movement of the handlebars becomes very apparent if you want to speed up your video. Mounting a camera directly to the bike anywhere picks up absolutely every mechanical bike sound from the shifters to the chain going through the rear derailleur.

If you don't have suspension the video can be shakey with weird vertical squishing. Wrapping a piece of old tube around the bar might help. At least with the chest mount your body acts as a shock absorber for the camera.

Let me say that on a wheel is not the best place:

Downtube looks kind of neat:

Skewer:
https://www.youtube.com/watch?v=W7jfcWEkSrI

I've been watching this guy's videos, and have learned some important things about group rides. but I see what you mean, there's a lot of shaking.

But this looks pretty good too. You get the full view, and it doesn't bounce very much, provided the road doesn't suck.