How important are my grades to the rest of my PhD career?
As far as your research stature is concerned, grades would matter least of all, below other non-academic stuff as your soft-skills, your personality etc. There are a number of reasons for this:


No one cares about your GPA once you are a researcher! While it certainly looks nice to have a stellar GPA, it's the work that you do and where you publish that would matter. Look up some resumes of notable faculty in your field - how many even list their MS/BS grades?
I might even say that your adviser would not be too happy if you have a 4.0 GPA - as it means you are spending time on perfecting your grades which is more profitably spent on research! I actually have read this on a faculty/university webpage - would post the link once I dig it up. EDIT: Haven't found the faculty website link yet, but here's a couple from an established source, phdcomics - enjoy!


PS: I'm actually waiting for JeffE to comment on this: His credentials are such that I'm not even qualified to state them, and he himself claims that he had the lowest undergraduate GPA amongst any professor he's ever met! 

EDIT: I'm talking about grades in subjects that you have to take as part of requirements of grad school - you would be expected to master the subjects that are directly required in your Ph.D research thesis, so your adviser would expect great grades in them naturally!

tl;dr - They don't matter (as long as you clear all your subjects!)