You notice that there is a tremendous developmental difference between a 4 year old and a 6 year old, on average. Similar differences occur in graduate school. While children spend their first 1 year coming to grips with fundamental issues of motion and eating, they develop speech in the next year and grammar soon after. After a considerable waiting period, they are then ready to learn reading and basic math. Grad students spend (hopefully no more than) a year on foundations, making sure they know basic concepts. Optimally during that time, but at least right after finishing minimum reuirements, they should work out in what area they want to specialize. This is probably the most difficult question they must answer, more so than the advisor quesiton. The reason is that "what do I want to study" can only answered by introspection, while "who should be my advisor" can be determined going down a pretty simple checklist.
Also, I should point out that students should be doing "the rounds" and talk to people even before they really look for advisors. It helps in all sorts of ways to know professors well. Eventually, many of these people will become colleagues: a current tenure track professor is as much in your peer group as 1 first year student is in the peer group of a finishing student!
Having chosen area and advisor it is imperative that you approach your prospective advisor for mutual screening. This should be in any case as soon as humanly possible, because delay simply shortens you learning span for the "reading and writing" equivalent: to conduct research on your own, and become a member of the math community in your area. In many cases known to me that lead to nice theses and nice jobs, this meeting was early in the second year of grad school.
If you dream comes true and you are accepted by your chosen math god, all your future math activity will be determined by the advisor, so I need not and cannot spell out how things proceed. However, here are some pointers how to find your area, and once that is found, how to pick a prospective advisor:
Area:
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If you have no particularly favorite subject, and haven't
spent time reading about areas of mathematics while working for
qualifiers (but you should have!!!) ask yourself:
* what do I dislike? (this eliminates a bunch of possibilities for
most people)
* what courses did I like?
This should narrow things down sufficiently to no more than 10-15
professors in your department.
Advisor:
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Your future advisor should
* have the reputation of a reasonable person with the senior graduate
students
* have published high quality articles in the last 10 years
* work in one of the areas that remain on your list
* not to be about to retire
* not be on sabbatical next year
* not frighten you by sight.
Make sure you talk to grad students that already have advisors and discuss with them pitfalls, experiences, etc. Besides this, it is a bit risky to choose people not yet tenured, (because they still have to worry about themselves), or worse yet non-tenured professors with small children.
This should narrow it down to less than 5, and hopefully more than zero. Go and talk to these people. If you decision feels like it's a risky choice, don't make it.
But make it early.
Grad school is not a spectator sport. You need to take it by the scruff of the neck and wring out the best possible outcome for you!