Hosting companies are eager to make you upgrade to a more expansive plan, but it's not always necessary. Do you use caching in your blog?
A dedicated server or a VPS (which I guess you mean) are different in performance (the "real" server being faster), but in both cases, you'll have to manage them yourself or pay someone to do it. Your host will probably offer managed servers, too, that's basically a machine that is managed by them so you don't have to worry about configuration, updates etc, and you're the only one on that machine. Wether that makes sense depends on how heavy each page view on your site is to the server and how many page views you get on a given day.