Hello Noah's Dad,
Great question by the way and I hope this answer will assist you.
1. I'd like to know how many people click play on a video when they are on a page that has a video.
You can use Event Tracking to do this (see below), but you need to make sure you have set up tracking for your website. More information can be seen here: http://code.google.com/apis/analytics/docs/tracking/asyncTracking.html
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I believe you can wrap the link around your video embed code and it will do the desired action of tracking people who played the video.
_2. I'd like to know how many people are clicking "like" " google plus, etc.) _
This one is a bit trickier. Most of the social media "like" or "+1" buttons are loaded / served up using iframes (for Facebook) or JavaScript for the others. Google provides its own markup tags to load their social media buttons in, and its all a bit of a dogs dinner quite frankly. There should really be a standard for the implementation of these buttons but anyway! You could try using a similar onClick function as shown above, but you would need some additional code in the footer of your site to attach the onClick event to each button after the page has been loaded. I'm not sure if there is an easier way to do this but if you need help with coding this up let me know. Equally if someone else here has a better method I would like to know too!
3. I'd like to know the path people are taking on our site. For instance, if they click a link from Facebook, and go to a landing page, what page are they visiting next
This is possible in Google Analytics. If your on your Google Dashboard for the domain, click Audience then click Visitors Flow, this provides a nice interface to view this information.