Dealing with Generating Functions accurately I'm currently working through Iven Niven's "Mathematics of Choice." In the chapter on Generating Functions, the exercises include problems like:
How many solutions in non-negative integers does the equation $2x+3y+7z+9r=20$ have?
This of course ends up being the same as finding the coefficient of $x^{20}$ in $(1+x^2+x^4+x^6+\cdots+x^{20})(1+x^3+x^6+\cdots+x^{18})(1+x^7+x^{14})(1+x^9+x^{18})$. Maybe I'm just being a big weenie, but dealing with polynomials this large ends up being really tedious and error-prone. 
I've realized that you can save the most involved multiplication for last, since that requires the least detail, and I'm of course not bothering to keep track of any powers greater than the one I'm interested in. 
Any tips for keeping these giant polynomials manageable?

When you can only reasonably evaluate by multiplying out the polynomials, you've described what is pretty much the only approach.  Just multiply out, simplest factors first, and keep dropping all coefficients higher than the one you are interested in.  In some cases this is all you can do, unfortunately.  (Or, use a CAS to do the messy algebra for you.)
However, occasionally you can simplify your generating function in a way that allows you to compute the coefficients by other methods, and this is sometimes (though not always) easier.
Let's take your case as an example.  The four polynomials in your product can be written as $\displaystyle \frac{1-x^{22}}{1-x^2}$, $\displaystyle \frac{1-x^{21}}{1-x^3}$, $\displaystyle \frac{1-x^{21}}{1-x^7}$, and $\displaystyle \frac{1-x^{27}}{1-x^9}$.  Because we don't care about what happens to the coefficients above $x^{20}$, we can ignore the powers higher than $x^{20}$, and take the product of the resulting functions.  That is, your problem is to find the coefficient of $x^{20}$ in $$g(x) = \frac{1}{(1-x^2)(1-x^3)(1-x^7)(1-x^9)}.$$
One could then use partial fractions to decompose the product into the sum of functions for which one already knows a formula for the $n$-th coefficient; this is the way to get Binet's formula for the Fibonacci numbers from the generating function $\displaystyle f(x) = \frac{x}{1-x-x^2}$.  For this problem that would be a lot of work, perhaps more than just multiplying the polynomials, but in general this is quite powerful. Another method is to take many derivatives, observing that the coefficient of $x^n$ is $\frac{1}{n!}$ times the $n$-th derivative at $0$, though in this case I expect this will be at least as messy as doing the original multiplication.
The full power of generating functions is not generally expressed in finding just a single coefficient, but in the way that it allows you to compactly work with all coefficients: e.g., the same generating function $g(x)$ above will give you the solution (via the coefficient of $x^n$) to this problem when $20$ is replaced by any integer $n$.
You may also be interested in this question and its answers.  If you are interested in learning a lot more about generating functions, Wilf's Generatingfunctionology is the way to go.
