I want to know what techniques are known to present a diffeomorphism on a surface with boundary (the diffeomorphism is not necessarily the identity restricted to the boundary) as product of Dehn twists (of course, up to isotopy). I know there are a few books I can look at but can someone briefly describe the methods, or at least refer to exact pages of a book? Perhaps this is not easy to do in general, can you choose a diffeomorphism and show me how you present it as Dehn twists on a low genus surface?

$\newcommand{\RR}{\mathbb{R}}$Dehn twists are orientation preserving, so you can never write a reflection as a product of Dehn twists. On the other hand writing an orientation preserving, periodic element $\phi$ as a product of Dehn twists is a nontrivial problem  that is to say, there are mechanical ways to get the desired product, but often the number of Dehn twists required is surprisingly large! For a slick (read, comprehensible) solution we need more details about how the surface and periodic element are given. In general, the mechanical solution is as follows.
Sometimes a slick solution can be found using the braid relation and its generalization, the "chain relation". In your comment, it sounds like you are drawing the surface in $\RR^3$ and the periodic element $\phi$ is induced by a rotation of $\RR^3$, so I will just point out the following nice fact, given as an exercise in Thurston's book: not every periodic element is induced by a symmetric embedding of the surface into $\RR^3$. Consider the element of order six in the mapping class group of the torus... 

