Another quadratic Diophantine equation: How do I proceed? How would I find all the fundamental solutions of the Pell-like equation
$x^2-10y^2=9$
I've swapped out the original problem from this question for a couple reasons.  I already know the solution to this problem, which comes from http://mathworld.wolfram.com/PellEquation.html.  The site gives 3 fundamental solutions and how to obtain more, but does not explain how to find such fundamental solutions.  Problems such as this have plagued me for a while now.  I was hoping with a known solution, it would be possible for answers to go into more detail without spoiling anything.
In an attempt to be able to figure out such problems, I've tried websites, I've tried some of my and my brother's old textbooks as well as checking out 2 books from the library in an attempt to find an answer or to understand previous answers.
I've always considered myself to be good in math (until I found this site...).  Still, judging from what I've seen, it might not be easy trying to explain it so I can understand it.  I will be attaching a bounty to this question to at least encourage people to try.  I do intend to use a computer to solve this problem and if I have solved problems such as $x^2-61y^2=1$, which will take forever unless you know to look at the convergents of $\sqrt{61}$.
Preferably, I would like to understand what I'm doing and why, but failing that will settle for being able to duplicate the methodology.

You can type it into Dario Alpern's solver and tick the "step-by-step" button to see a detailed solution. 
EDIT: I'm a little puzzled by Wolfram's three fundamental solutions, $(7,2)$, $(13,4)$, and $(57,18)$. It seems to me that there are two fundamental solutions, $(3,0)$ and $(7,2)$, and you can get everything else by combining those two with solutions $(19,6)$ of $x^2-10y^2=1$. Using mercio's formalism, $$(7-2\sqrt{10})(19+6\sqrt{10})=13+4\sqrt{10}$$ shows you how to get $(13,4)$; $$(3+0\sqrt{10})(19+6\sqrt{10})=57+18\sqrt{10}$$ shows you how to get $(57,18)$.   
