I'm surprised, reading the various answers and comments to this question, how much support there is for the idea of reading EGA. It is evidently a mathematical masterpiece of a certain kind, but I would never recommend it to a student to study.
As to how much time to spend on EGA and SGA, this is something that you ultimately have to decide for yourself, hopefully with the guidance of your thesis adviser. But one thing to remember is that many very clever people have pored over the details of EGA and SGA for many years now, and it so it is going to be hard for anyone to find interesting new results that can be obtained just by applying the ideas from these sources alone (as important as those ideas are). Even if you want to make progress in a very general, abstract setting, you will need ideas to come from somewhere, motivated perhaps by some new phenomenon you observe in geometry, or number theory, or arithmetic, or … .
This is part of the reason why I advise against spending too much time just holed up with EGA and SGA. By themselves, they are not likely (for most people) to provide the inspiration for new results. On the other hand, when you are trying to prove your theorems, you might well find techincal tools in them which are very helpful, so it is useful to have some sense of what is in them and what sort of tools they provide. But you will likely have to find your inspiration elsewhere.
I think it is worth thinking about two of the most significant recent theorems in algebraic geometry: the deformation invariance of plurigenera for varieties of general type, proved by Siu, and the finite generation of the canoncial ring for varieties of general type, proved by Birkar, Cascini, Hacon, and McKernan, and independently by Siu.
Siu's methods are analytic; one might summarize them as $L^2$-methods. I'm not sure what text one would begin with to learn these methods. Certainly Griffiths and Harris for the very basics, but then ... ? The methods of BCHM are techniques of projective and birational geometry. A careful reading of Hartshorne, especially the last two chapters, would be a good preparation for entering the research literature in this subject, I think.
It's not clear to me that one gains more essential background by reading EGA.
An aside: there is much more to EGA than just handling non-Noetherian schemes, but the spectre of non-Noetherian situations seems to loom a little large over this discussion. Thus it seems worthwhile to mention that, while non-Noetherian schemes arise naturally in certain contexts (as Kevin Buzzard noted in a comment on David Levahi's answer), I think these contexts are pretty uncommon unless one is doing a certain style of arithmetic geometry. Somewhat more generally, I don't think that flat descent should be the focus for most students when learning algebraic geometry. It is a basic foundational technique, but I don't expect most interesting new work in algebraic geometry to occur in the foundations. (More generally still, this is probably a good summary for my case against spending time reading EGA.)