For tablet and other mobile devices, new browsers are popping out like mushrooms, and it's practically becoming very difficult to test the rendering of every website we design on all such browsers. Is it a good idea to use Normalize.CSS on every website?
I like to use Normalize.css whenever I possibly can on a new project. It's better than Eric Meyer's CSS Reset in that it actually provides "standardized" styles for elements, as opposed to removing all styles.
So yes, it's a great starting point for a new project. You'll want to ensure that the framework or CMS your using won't conflict with it, but it's a great resource.
It's just a tool. You need to evaluate whether or not it's appropriate to use for yourself. For any given project, you might be using some other elements that conflict with it, for all we know. It's a handy library, but nobody can really make a blanket statement that you should/not use it for every project you do.