Actually if you are just beginning, you can take many different routes. First of all, if you are wanting to start one just to see how you do it, and it is not going to be something very serious, it does not cost a thing. You can use a website like www.freewebs.com, or geocities.yahoo.com. Those can be completely free, and have a very nice starter in there for you.
That is the beginner way to do it, and is probably one of the bets ways to start. Another thing I advise you to do is to learn a bit of html. There are several tutorial sites on this, and I advise you to look into those to try and learn some of that. It will begin by looking very complicated, but just take it in stages and it will go by nicely. A site I use all the time for it is www.w3schools.com.
Now lastly, if you actually get into this big time, you can get much more involved with it, and start a great looking site. You will probably want to buy a website creating program, for example: Dreamweaver, or FrontPage... You will then want to buy yourself a domain, which in itself is not overly too expensive. After you upload everything onto this site, then you will need to get it hosted. If you do it correctly, you can usually get a really good deal from the people that you bought the domain off of too, and have them host it for free if you bought it in a package, or for very cheap.
So to answer your question in something short, if you want to just start one and have a basic site to just have something up, it costs nothing more than time. If you want something more professional looking, and something that you will want to continuously update and have this be public, then it will begin to cost money. Also costs involved could be a template if you want to have a set for your site. So for costs, you want to look into a Domain, hosting, templates, website editing program, and your time.
Good luck with your website, and I hope this helps :) Feel free to e-mail me with any other questions.
--Lance