Can I use Ravi Vakil's way of learning for elementary subjects? I mean you can't learn math in a linear order. Can I just read a paper first on a subject I haven't studied and just work backwards? For example, I have never studied combinatorics but I sort of have a fuzzy idea about the paper on alternating permutations by Stanley. 

This may in the end, for an absolute beginner, be quite unproductive.  Normally introductory texts are written in such a way as to be helpful to a newcomer by first building a knowledge-base of basic facts about a field of mathematics, and then perhaps having a few chapters on selected topics of moderate difficulty towards the back.
I would recommend a compromise.  First pick up an elementary text which is well-known and accepted among your peers, or accepted among the people who are writing the papers you wish to read.  (This information is not very hard to find.)  Then read and study the first few chapters with the basic material thoroughly.  Then you may read the advanced paper, and be thoroughly confused, and work your way backwards as you describe.
But without the first step I do believe you will be not only creating more problems for yourself in the beginning, but also in the end, where you may indeed miss the entire point of a paper.  Very often top research papers replace a standard part of a big elementary machine with a clever alternative to attack a difficult problem and find a new solution.  You will not understand or appreciate this without a basic knowledge of how the solution works in the standard case first.
