The general rule is that bike components are a lot more expensive bought piecemeal than when bought as a part of a bike, so an overall upgrade is usually poor economics. It can be fun, and it can be made to work if you're either very good at bargain hunting or have a serious stash of parts at your disposal. But this is all assuming that you have a good quality and good fitting frame to lavish all your attention on to start with.

So, the sensible course of action is to replace-with-nicer once something breaks or wear out, and to change those things actually needed for you to be able to use the bike well. Things like saddle, stem, bar, pedals, tires etc.
But for upgrades of things that already work you're better off saving that money for another bike further on. If you keep riding you'll want to have a beater bike around anyhow.