I don't think that a Facebook photo page alone will do much. Facebook's revenue model involves only showing posts to a subset of your potential audience — even those who have "Liked" your page, unless you pay for promotion. Paying to promote posts of photos you really want others to see might get you a bigger audience, but even then... uphill battle.

There is an overwhelming amount of photography online. Your work is going to have to be particularly novel to get attention. Not just technically good, or artistically done — really eye-catching. And by definition, there's a 99.9999% chance that that's not you. I don't mean that in a negative way, either... the best photography isn't necessarily the most gimmicky, but having schtick is basically going to be necessary in order to get attention online — some hook to "go viral".

Here's what I'd do: concentrate on your own web page. Update it more frequently with your best work — but only your best work. (You'll be judged by the outliers — if you have something really amazing, people will count that, but they'll also remember the worst.) But don't focus on promoting that per se. Instead, make some prints — again, of your best work — and get involved in the local community. Go to art fairs, street festivals; show off at open studios if your area has that kind of thing. If you're into portraiture, maybe give coupons for discounted sessions. Build up a following and word of mouth, and people will naturally look for your web site.

Alternately, if building up in that way isn't really what you're looking for, I suggest an online photo community specifically for the purpose connecting around photography. 500px, Flickr, etc. Post your stuff — but also get involved in commenting on the works of others, contents, and so on. You're really unlikely to break out of the crowd with this approach, but maybe that's not what you want. It will get more people looking at and interacting with your work.