While having a Facebook page is a great idea and I think there is no reason to not set one up, having a dedicated website for your photography is ideal. You have the option of making your own site by using open-source gallery software and installing it on web hosting, using a gallery site like Flickr/Google Picasa Web, or using a service dedicated to serious photographer galleries. Even if you know how to do computer programming, I would strongly suggest using one of the services geared toward dedicated photographers out there because they offer a few special features you can’t easily get elsewhere. With popular services like Zenfolio and Smugmug, they allow you to have a fast dedicated image gallery with smooth well thought out user interfaces that top completely free solutions.
A web-based image gallery using Zenfolio.
If you opt to pay a reasonable yearly fee, you will have the ability to sell prints and downloads of your work, which I think is a great feature and reason enough to use one of these rather than rolling your own solution. You will also most likely be able to upload an unlimited amount of photos, offer password protected galleries, and supplement the gallery with information pages allowing the gallery to be a one stop shop.
My Zenfolio based site has been a pleasure to work with, while not perfect (I use Linux and options for uploading large numbers of photos successfully in Linux is limited as of this writing…), it has been a great purchase overall. I have even had a few orders from cosplayers since I transitioned from a community gallery site called Flickr to my own dedicated site on Zenfolio. While sites like Zenfolio and Smugmug usually offer a free account, their paid accounts allow you to link up a domain name of your choice to the service. You can register a domain on any of the multitude of registration sites out there and easily link that up with your gallery hosting account (just make sure the service you choose allows for it). Having a professional level gallery with a dedicated domain instead of site.zenfolio.com or site.smugmug.com shows people you are serious about your work.