Yes, that's right.
If you can be bothered.
build and design the mobile version first and then work my way outward to the tablet, netbook and desktop sizes afterwards? (To follow best design practice)
It's just one option, but not the only way. I personally build each 'version' at the same time. Depends on the site, I guess. Some would argue that its best to start with a basic, very simple layout that will work on all browsers, and then add in media queries for browsers that can understand them. From that perspective, even old version of iE will be fine, as they'll get the basic styles.
Well, it depends on what the basic design is. Responsive web design is actually not a new thing. It's been around since the beginning of the web. The first websites—built just with HTML—were fully responsive, because HTML is responsive! It's only CSS practices since then (like fixed widths) that have prevented sites from being responsive. So really the title of this should be "responsive CSS" ... as this is all about styling sites in such a way that they can expand and flex depending on the viewport size.
You can build a fully responsive site without @media rules at all. Just avoid fixed widths on your wrappers etc.
are there old Safari / Firefox / Chrome / Opera version that do not support it either?
If you go backfar enough, they won't support them, but it doesn't really matter, as these browsers tend to auto update, so few people are using older versions. Again, though, if you have basic fallback styles, then any site will look decent, no matter how old the browser.
Perhaps I should be taking a much closer look at the mobile browsers too...
It's only really the smartest smart phones that really support @media. The rest can just live with the basic fallback styles you create first up. Here's a good resource on that: