Is it neccessary to study things earlier? I got confirmed from a graduate school starting from next year and I will major algebraic geometry.
Until now, I have never thought that I study little things than others with my age. However, I heard that some of my colleagues already studied Hartshorne at least once and quite a few of them have read Rudin's RCA when they were undergaduates. It's kinda unbelievable to me, but it seems like if they really did study and understood, then they will write absolutely a better Ph.D thesis than mine.
So I'm now very worrying myself. I want to know whether this situation is general. Is it recommenable to study graduate subjects as early as possible? Or are there people here who experienced the same thing too? Was that beneficial?
Between "studying each thing deep and slow" and "skimming many subjects as fast as possible", which one is better?
 A: Your questions in order:
Yes, it is highly advantageous to be exposed to more sophisticated ("graduate") subjects as early as one can tolerate it.
Not clear that one should "study" them.
Yes, some people do "read ahead". I myself found it very helpful.
I would claim that the widely-believed sense of "deep and slow" versus "skimming ..." is a fake comparison, and is not the question anyone should really ask. That is, studying "slowly" cannot possibly be "deep", because "slowly" also entails maintaining naivete and one sort of shallowness for an unfortunately long time.
Likewise, the "skimming... as fast as possible..." is not any sort of "other" alternative. A more sane "other" is "looking around, not getting bogged down in details, trying to see where things are going".
And, my recommendation would be to both read fairly carefully and also look around to see what's going on. Both lower-level details and some idea of the goals and larger enterprise.
Very specifically: much of "comfort" and "facility" consists of familiarity more than anything else. The psychological obstacle of novelty is surprisingly great, while the psychological ease of (even superficial) familiarity is surprising. Simply hearing the words and a bit of a story a year or two (or more) earlier is extremely beneficial, in all my observation (and my own experience).
A: Your question is very broad, and I'm not sure this fully addresses it; but this is too long for a comment, and hopefully you find it useful nonetheless.

I think there's a couple false assumptions here.
First, that there is a "better" way to approach studying mathematics. People vary wildly in how they learn, and ultimately I think it's best to find an approach to math that works well for you. Certainly you shouldn't discount the value of mastering difficult material early (and although rare, it is definitely believable for an undergrad to master such material), but at the same time, just knowing a bunch of stuff doesn't make you a mathematician. Grad school is much more about learning to be a mathematician than it is about learning material.
Second, and far more importantly, your line

It seems like if they really did study and understood, then they will write absolutely a better Ph.D thesis than mine.

This is something I'm still struggling to understand intuitively, so saying this in answer to your question also helps me internalize it: mathematics is not linear. Even ignoring the fact mentioned above that mathematics $\not=$ a bunch of facts, and even ignoring the fact that one's rate of learning changes over time, it is impossible to guess ahead of time how your thesis will compare with someone else's simply because there are so many different facets of mathematics. As you go through grad school, regardless of where you start relative to your peers, you will eventually find yourself an expert in some small area, just as they will in their own small areas. What contributions you make to this small area will surprise you, and it's pointless to try to guess ahead of time whether they will "match up" (however you might measure that) with someone else's. 
Your thesis is not predetermined; it will be the product a number of things, including the growth you experience as a mathematician as you go through grad school (as well as a fair amount of chance, let's be honest). Certainly knowing more things at the beginning is an advantage, but it's by no means a dispositive one.
