As other people have said, to get your own website, you find a hosting company and buy a plan from them. $5/month is about par, and watch out for hidden fees beyond the month price and price of the domain name. You should be able to get a year's service for less than $100.
To work on yoru website, you use FTP. You write files in Notepad or some other plain-text editor, like Wordpad or Simpletext, and you use exactly the same HTML you use on your free websites, only you're not forced to include ad banners. Be sure to include all parts of the pages. An FTP program should come with the hosting plan, and the way mine works is I just drag my finished filed from my computer folder into my website folder. If you already know HTML, and the service comes with some FAQ, you shouldn't have any problems. Good luck!
Which hosting company do you use?
I'm flattered that you're asking me, but you should really ask everyone who has a website. I bought mark356.net from featureprice.com last November, but as well as $5 a month and $25 for a domain name, they had a $25 hidden fee. Just recently, they were bought by atlantic.net, and they didn't give atlantic.net the money from their cheap customers like me, so they're only giving me until the end of July to give them more money, and I'm not sure if I'll sign up with them again. However, mark356.net always seems to load well, and the FTP is as easy to use as I've described, and there is a good stats page.