If you want something, ask for it.
Seriously: you can use Facebook Authentication {instead of|in addition to} your site's registration/login system. It's really not that hard and it's well documented (pay attention to FB's data policies though: what you can do/must not do with the data, how long you can keep it etc.)
When users sign into your app through FB Auth, they must grant your page (temporary) access to their basic profile (at least, I haven't found the way to only use FB Auth for authentication, without granting access to profile data).
On the other hand, if you are planning to track your site's users on FB without their knowledge and/or consent, there's a word for that: "stalking"; in some places, there's even a penalty of law to go with it. In such case, I would recommend talking to a lawyer first - just out of curiosity ;)