To find out if your site is splitting domains, try going to different pages and seeing if they're served. For example, go to http://example.com, http://www.example.com, http://example.com/, http://www.example.com/. These should all end up with the same URL at the top when the page loads. If it doesn't, that means your server needs to be configured to do these canonical redirects. Depending on what software you use, this can vary, but the Q&A forum has had a lot of answers related to configuring redirects, so look there, or try Googling it.
I'm not sure it'll make a huge different if you decide to use example.com or www.example.com, but whichever you do choose, be consistent throughout your site.
If you really want to learn to get into the nitty-gritty of web pages, the main tool I use to view web requests, http headers, source, etc is Firebug, a Firefox Add-on. Chrome has something similar, which I think came pre-installed (if you can right click something, and choose "Inspect element", you have it). To learn it, play around with it a bit. The HTML tab in Firebug (Elements tab in Chrome) will show you the source of where you're selecting in the page. The other tab it sounds like you'll be interested in is the Net tab (Network tab in Chrome). Here you can see all the different files the page is loading, and you can view their statuses, headers, and responses.
Another tool I use frequently in Firefox & Chrome is the Web Developer Toolbar. It lets you enable/disable a lot of different things (caching, images, CSS, JavaScript), and clear your cache with a few clicks.