undergraduate math vs graduate math It's really a mild, soft question.

So far, I am an undergrad student, contemplating on several majors.

What will be the major difference if I become a math major and go to a graduate school to study math?

Will a graduate math major generally focus on reading/studying recent research papers? Or will  the graduate math major be taught by professors on the things not covered in undergrad programs? 

The reason why I am asking this question is that undergrad math programs seem to me at this point so well-covered that after studying the programs, a person will be able to do anything he wants to do, and if one wants to do research, he will be able to catch up with recent progress by reading research papers.

If this is true, why is the graduate school even needed?

By the way, I am a first-year undergraduate student :)
Here is what I tell my grad students:

The difference between undergrad mathematics and graduate mathematics is the difference between art history, or art appreciation, and learning to be an artist.

As an undergraduate you see a lot of mathematics, but you don't create new mathematics.  The goal of graduate school (and here I am speaking from experience with top fifty U.S. graduate schools, so what I am saying probably applies best in that context) is to learn how to create new mathematics, and then to create that new mathematics.   

One specific consequence of this (in my view) is the following: often in undergraduate mathematics classes, proofs and rigor are presented almost as moral imperatives --- as if it is a moral failing to know a statement without knowing why it is true; consequently, people often put a lot of effort into learning arguments just for the sake of having learnt them.   (This is exaggerated, perhaps, but I think it reflects something real.)  On the other hand, in research, one learns arguments for different reasons: to learn technique, to pick out important ideas --- there is a professional aspect to the way one looks at pieces of mathematics which is not usually present in undergraduate mathematics.  One gives proofs in order to be sure that one hasn't blundered; one's interaction with the mathematics and the arguments is much more visceral than in undergraduate courses.   

(I am not speaking from any experience now, but I think of the difference between learning how to interact with a block of marble, and bring a new form out of it, however rough it might be, in comparison to looking and learning about a lot of existing beautiful statues, masterpieces that they are.)
User2346:

Here's the opinion of someone who majored in math (at a top school in the United States), but was not a genius at it. 

This is a condensed, generalized, down-and-dirty opinion, but I stand by it.

Your assumption about being well-covered is more accurate that not. If your intent is to form a mathematical basis of strength that you can apply to applications of math (e.g. finance stat econ) then undergrad is as far as you need to go with it. By the time you take Real Analysis &amp; Complex analysis your mind will have developed in an amazing way. As Matt Pressland said, you learn how to learn Mathematics. Matt also mention the difference between that style of meta-cognition and being able to research. It's impossible to say today that you could or couldn't come up with a unique research idea that would be considered a contribution and that you're passionate about to pursue. However, unless the word research makes your heart throb, you don't need Math graduate school.

I've rewritten the rest of my answer to be a bit more compact.


Do not major in math, minor in it. You will gain a superior analytic mind. However, your employment potential depends on not being a Math major. The Corporate world (Actuarial aside) doesn't know wtf Math is. They will ask if you wanted to Teach High School.
Accounting is boring, and Economics is fluffy. Choose Statistics and augment with Finance. Also, Computer Science requires more hours spent at 1-sitting than Mathematics. Engineering trumps Computer Science as you can learn to be a programmer in your spare time.
First know what you want to do as a career. Do not decide this on your own. Go to your career center for expert help, and shadow professionals and intern. Funny how this is #3.


Conclusion
Math can be a minor. Let your possible career path determine your major. At the least you can be earning a salary when you're 22-23 while deciding what new career you want. At the best, you'll save yourself 3-5 years of searching, finding-professional-self, and rebuilding qualifications.

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I decided to add some external links that would shed light on why some of my views are as they are. These come fro the Bureau of Labor Statistics... Oh look Statistics is even in the name =)

A small summary is included with the link, but please click and compare. 
These are just samples, you can find your own BLS data to refute if you want.

Occupations from mostly-pure discipline

1. Economist. $43/hr. 6% projected growth. 15.4K jobs. Bachelors Entry-level
http://www.bls.gov/ooh/Life-Physical-and-Social-Science/Economists.htm

2. Mathematician. $48/hr. 16% projected growth. 3.1K jobs. Masters Entry-level
http://www.bls.gov/ooh/math/mathematicians.htm

3. Statistician. $35/hr. 14% projected growth. 21.5K jobs. Masters Entry-level
http://www.bls.gov/ooh/Math/Statisticians.htm


Occupations heavy in STAT with Finance Augment

4. Financial Analyst. $35/hr. 26% projected growth. 236.0K jobs. Bachelors Entry-level
http://www.bls.gov/ooh/business-and-financial/financial-analysts.htm

5. Actuary. $42/hr. 27% projected growth. 21.7k jobs. Bachelors Entry-level
http://www.bls.gov/ooh/math/actuaries.htm




Conclusion for the sample

Stat has way more job positions filled/needed much faster growth than Econ, same entry-level as Math.

Finance augmented,  Statistically heavy jobs are quite abundant, pay well, and grow fast. This is what I recommended. I recommended majoring in Stat with a Finance augment. Both examples of this require only a bachelors and are in a job space projected to grow much more than the Mathematician or the Economist. The pure Statistician indeed makes less than those two. So to repeat from previously: Math Masters not needed if you like other things. Don't major in Math in undergrad.