While you certainly need to be aware of HTML5 and the new elements it brings, I'd say it's much more important to focus on learning CSS3 which adds so many new controls for presentation, especially as it relates to mobile devices. As you do, you'll also need to keep an eye on browser compatibility issues. We still have to contend with a significant number of users who persist in using old versions of Internet Explorer, so you need to know which controls work in which browsers as you create new websites. Sometimes it's critical, and sometimes it's not.
The fundamentals of HTML and CSS do not change with HTML 5, so that's where you need to start.
Quite apart from the new features, it is worthwhile declaring your web pages as HTML5 because it does not allow you to use obsolete code like <font>. So you will learn how to do it correctly from the start.
It's not like there's this tremendous difference between HTML 4 and HTML 5, or CSS2 and CSS3. These are evolutionary steps and if you start with the older versions, it's not a giant leap to the next generation. But, as I said, in the current environment, it does make sense to be aware of the differences in HTML5 so you'll create webpages that can be more easily transitioned later, and to learn CSS3 (which works in both versions of HTML as long as the browser supports the feature involved) so you can take advantage of the widest range of design options.
Until very recently, I was one of the people who said to stick with HTML4. Then I decided to convert one of my websites to HTML5, just as a test. It was so easy, that I converted the lot. As rtrethewey has said, the difference is incremental, so you can use either. I just found no good reason NOT to use HTML5.
Except perhaps the fact that the otherwise excellent WDG HTML validator is not HTML5 aware. So it flags up numerous "errors" that are, in fact, correct in HTML5. I have found a solution by front-ending the W3C HTML validator to make it more useable. When time permits, I'm working on a version that combines the HTML5 validator with a CSS3 validator, and a broken links checker all in one.