I'm in the same boat as you are - I'm looking to start a website for the Non-profit group I work for. While it won't be a ton of help, I can probably give you a couple of places to start.
Basically, in order to start a website you need two things - a domain name (web address) and a web host. I'm assuming you weren't looking to host your own website (run the website from your own computer or a secondary one). Some companies will offer both web hosting and a domain name (Like Godaddy) but a google search will turn up a load of different options. Ipage seems to be a popular one for those who don't have much experience in web design. However, many of the sites offering easy to use or "drag and drop" web building tools, don't look quite as professional as some nicer web pages I have visited.
My suggestion - if your looking for a really well made website, something professional for a business or similar, I would either find someone who can help you get one started and show you how to maintain it (I might go that route

), or look into having one designed and hosted for you (Godaddy offers that option though it's much more expensive - 50 bucks a month if I remember right). That way, you get a nice website and you won't have to worry about running into problems with server issues, downtime, etc. I have never gone that route though, so I don't know too much about the end result.
I've been looking into designing my own (to save money :thumbup

- I'm probably going to register a domain and host it through Wordpress. I just recently found out that they offer hosting and even though it's more blogging orientated, I can still design one that looks more traditional (if that's the right word for it). There's a lot of information on how to start a website, how to learn HTML coding, uploading pre-made websites, etc. There's also many different programs and places to look into for web-design - Dreamweaver seems to be very popular as a paid program. I would look at Drupal, Joomla and Wordpress too for open-source options.
Here's a few links for a bit more info on everything...
http://www.thesitewizard.com/gettingstarted/startwebsite.shtml
http://support.hostgator.com/articles/hosting-guide/starting-a-website-for-beginners
Hope this helps and good luck!