Component upgrades seldom make economic sense. I think that for a project like this it's important to think it through and cost everything out before you get started. You may find the total cost to approach the price of a whole new bike.

Everything on a bike works together so you can seldom replace just one part without having to change something else to make it work. The big money is in the shifters so that pretty much drives everything else. 10-speed shifters means you'll need a 10-speed cassette and chain. The rear derailleur will work but I suspect that's one of the parts you'd like to change anyway. I've never tried to upgrade from 8 to 10 speeds so I can't say if your existing crankset and front derailleur will work. I know that Shimano doesn't recommend using a 10-speed crankset with non matching components so I assume there's a dimensional difference somewhere.