I would like to monitor users' page visits and clicks in my Rails app to make recommendations. My questions are:
There are other ways, but the vast majority probably use Google Analytics. Several gems exist that help you integrate with GA to get at the data. See here: https://www.ruby-toolbox.com/categories/Web_Analytics.
Based on your first question, it seems you may want more insight than GA can provide. I've used ClickTale (http://www.clicktale.com) and Woopra (http://www.woopra.com) before, to good effect. This article lists several other alternatives, too - notice the high marks for Clicky: http://imimpact.com/web-stats-alternatives-to-google-analytics/.
Google Analytics (and almost all of these others) will take care of your second question automatically whenever the user loads a new page, since it keyed by URL. That means that, although you put the GA script code in a single place, each unique page is tracked individually.
If you have AJAX requests that change that page without changing the URL, you'll need to dig in to the GA script API. Essentially you'll need to push a new url (possibly with a # in it) whenever you want to track an AJAX-driven link/button click. See here: http://davidwalsh.name/ajax-analytics