I'd break it down into parts and consider each and your interest (and suitability) for each and go from there. Starting out is starting out and a hobby is a hobby, but if this is something you are interesting in pursuing as a career or something...
Business and marketing: Websites need real objectives, consume resources and cost money. Websites tie into overall marketing programs and in turn advertising, social media initiatives, event promotion, etc. I'm just mentioning these considerations here, they are a world unto themselves, but it's the reason someone would pay you to sit in your double wide trailer in your underwear smoking weed using a cracked copy of Dreamweaver on a Netbook you got for xmas.
IA/UX/UI: I'm grossly lumping these together and you can just read the wiki but each of these is a discipline unto itself:
http://en.wikipedia.org/wiki/Information_architecture
http://en.wikipedia.org/wiki/User_experience_design
http://en.wikipedia.org/wiki/User_interface
Graphic design is what it is. Important. Subjective. More talent than hard work in my experience. You can get passable with hard work but raw talent will ALWAYS wipe the floor with you.
Ok, maybe what the original question was about. Building websites.
I'd start with the dorky crap. I'd build a freaking web server. For free on a junky older computer, for cheap with used craigslist parts, for cheap with low spec parts like Athlon II or Core 2 Duo. Could build a new one for $200-$300? Or, if you have enough ram install something like vmWare Fusion and try out openSUSE (my longtime fave linux) or I just tried out Ubuntu server and am loving it, but it's all CLI, no GUI. Anyhow, the reason to do this is to get a better understanding of the whole process. Screwing with network cards and hostnames and getting beat up by permissions issues will make you undertand what is really going on. I installed ubuntu server and had LAMP up and running, my IP set to static, had phpMyAdmin installed and mySQL configured with a user I could use to access the database, installed subversion and an ftp server and got it up and running in about 1 hour total. When I want to develop I fire up my VM and I have a full apache2 server with php and mySQL and a fully functioning .svn server. I prefer this to using the stuff built into Snow Leopard, but that is yet another option. OS X has an Apache server installed. PHP is installed but not activated and it's not to hard to get mySQL up and going. Subversion I had less luck with but there are tons of host with free .svn hosting. Chefs do this when they go to the farm to see where the food comes from. It just gives you a real understanding to start at the very base, even though learning Linux will make you scream at times.
Rackspace has cloud servers with root access for $11/mo.
http://www.rackspacecloud.com/cloud_hosting_products/servers/
Learn subversion or Git.
I like these guys for good free accounts:
http://www.assembla.com/
I used to program in Director then that went away and then I start doing a lot of Flash and now... I'm back to basics with HTML, CSS, Javascript and stuff and mobile is a fact of life now as well. Websites have to work on mobile now. I used to use ColdFusion but found it too hard to sell to clients and IT people and now I use PHP.
Point is, specific skills will come and go, but you need to learn how to learn, learn what to learn and learn what to start moving away from. It can be overwhelming.
Right now, I'd learn HTML5, CSS3 and javascript libraries like modernizr and jquery. I'd learn the newer stuff but learn how to fall back to the older technologies like HTML, XHTML and CSS. I would learn about responsive design and theories on how to deal with mobile and tablets.
http://www.alistapart.com/articles/responsive-web-design/
http://www.modernizr.com/
http://jquery.com/
http://www.jqtouch.com/
http://www.sencha.com/
http://www.phonegap.com/
Learn to make pages go from 27" Cinema to iPhone.
Sucks but I'd learn PHP and mySQL. Build small things. A list maker. A CRUD system to store your DVD collection with links to IMDB. Play with Twitter APIs. Consume data from web service. Hell, convert your DVD collection to work as a web service and consume it from a small mobile phone web page app. Databases are going to the cloud, but mySQL is still good to learn the concepts on.
Just get on your homemade web server and play.