In a way the second item (register a domain name) is more immediate, in that you want to do that as soon as possible, even if you’re not ready to put up a site; they tend to get snapped up when you’re not looking.
Look on sites like c/net for reviews of hosting providers. National level providers offer basic shared web hosting for $10 a month or less, if paid for by the year, sometimes with a modest $15) “setup fee”. You can also pay for most by the month or quarterly, at a slightly higher cost, which may be worthwhile if you are unsure about the host, you can always opt for yearly payments later.
Basic “shared hosting” if fine. You don’t need anything fancy, or a business account to start. Given a choice between “Windows Hosting” and “Linux Hosting” or “Unix Hosting” (the type of operating system on the server where your site site hosted) I choose Linux or Unix, as it is often cheaper, I believe it to be more reliable and flexible (a persona bias) and the only advantage I see to Windows Hosting is that it enables the use of Microsoft Front Page, which I will strongly advise you against.
Extras – see if the options from the basic plan from one company appeal to you more than another. Do you want a blog as well as a web site? Do they offer the ability to host additional domains without charge (incase you want to have yourname.com and yournameillustration.com point to the same site)?
Templates and “site builders”. These have the appeal of promising to allow you to build a site with no web design or HTML knowledge. I don’t want to turn your off the to them out of hand, perhaps there are good ones out there, but my experience with these things is that they are limited and inflexible, and enough of a pain to learn to deal with that the same learning curve would give you a basic working knowledge of HTML, something you can apply anywhere, not just to one proprietary template system.