I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from text-books in advance, since trying to do so will lead you to being bogged down in books forever.
I think that Paul Siegel's suggestion is sensible. If you enjoy reading about different parts of math, then build in some time to your schedule for doing this. Especially if you feel that your work on your thesis problem is going nowhere, it can be good to take a break, and putting your problem aside to do some general reading is one way of doing that.
But one thing to bear in mind is that (despite the way it may appear) most problems are not solved by having mastery of a big machine that is then applied to the problem at hand. Rather, they typically reduce to concrete questions in linear algebra, calculus, or combinatorics. One part of the difficulty in solving a problem is finding this kind of reduction (this is where machines can sometimes be useful), so that the problem turns into something you can really solve. This usually takes time, not time reading texts, but time bashing your head against the question. One reason I mention this is that you probably
have more knowledge of the math you will need to solve your question than you think; the difficulty is figuring out how to apply that knowledge, which is something that comes with practice. (Ben Webster's advice along these lines is very good.)
One other thing: reading papers in the same field as your problem, as a clue to techniques for solving your problem, is often a good thing to do, and may be a compromise between working solely on your problem and reading for general knowledge.
Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up
applying R'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!
I didn't come anywhere near to reading the whole thing, but I did spend many hours
dipping into various portions while I was a graduate student. Serge Lang had put his copy into the
mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost
in thoughts or words. Even the bits I read of course were hard going. However, one thing was clear
even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite
a bit if some of the discontent could have a purely mathematical source.
A clue came shortly afterwards, when I heard from
Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: `Surely there
are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?'
Now I don't remember if it's in R&S as well, but I did read somewhere or hear from someone that
Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside
of the French school. Again, I have no opinion about the social aspect of such a sentiment (assuming the story true), but it is
interesting to speculate on the mathematical context.
There were in Orsay and Paris some tremendously powerful
people in arithmetic geometry.
Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his
writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together.
One suspects that the habits of the Grothendieck school,
whereby the six operations had to be established first in every situation
where a problem seemed worth solving, could be enormously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.]
Years later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has
changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well
with homotopy theory.
To summarize, I'm suggesting that the mathematical
content of Grothendieck's strong objection to motives was inextricably linked with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of
an obstruction)
in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points (the most basic maps between schemes!) in any non-abelian setting, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.
Correct or not, this overall impression was what I came away with
from the reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to
Diophantine geometry that used homotopy groups. Since I'm rather afraid of extremes, it was pleasant to find out eventually that
one had to go back and find some middle ground between the anabelian and motivic philosophies to get definite results.
This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)
Added, 14 December: I've thought about this question on and off since posting, and now I'm quite curious about the bit of R&S I was referring to, but I no longer have access to the book. So I wonder if someone knowledgeable could be troubled to give a brief summary of what it is Grothendieck really says there about the six operations. I do remember there was a lot, and this is a question of mathematical interest.
Best Answer
The question implies that there are pros and cons to reading classics, but only pros to reading the modern treatment. This seems contentious.
For instance 'The exposition may be presented in a cumbersome manner in view of modern machinery.' might seem to make sense, but it requires you to learn the modern machinery. Is the correct way to talk about orientability of a manifold to talk about homology with local coefficients? Maybe. If a beginner asks what orientability means, do you start by defining modules over group rings over fundamental groups? I think not.
Similarly, 'The terminology may be archaic.' is only a critique if you know all the modern terminology and none of the classical. Maybe Weyl's Riemann Surface book is useless to algebraic geometers of today, but I don't really know what a scheme is, so many modern books are useless to me (in fact, it often goes the other way, if there's some modern math I don't understand I try to find an old master who wrote before/during the time the terminology is being hashed out).