What are the basic affine transformations on a distribution for the various moments? To change a distribution's mean, we do the translation affine transformation. E.g. add a constant to every data point. 
To change a distribution's variance, we do the scale affine transformation. E.g. multiple every datapoint by a constant.
Are there similarly canonical affine transformations for kurtosis, skew, and high order moments? 
 A: Not affine, but polynomial transformations for example. Basically this is necessary because you want to change several numbers at the same time, and an affine transformation only gives you two coefficients to "play with" (I assume we're speaking about distributions on the real line).
If you consider a coordinate transformation $y=f(x)$, a density function $\mathrm{p}(x)$ changes in such a way that $\mathrm{p}(x)\,\mathrm{d}x = \mathrm{p}(y)\,\mathrm{d}y$, where $\mathrm{p}(y)$ is the density function in the new coordinates (related to the old one by a Jacobian determinant).
Denote the $n$th raw moment in $y$-coordinates as $m_n := \int y^n\,\mathrm{p}(y)\,\mathrm{d}y$, and in $x$-coordinates as $M_n := \int x^n\,\mathrm{p}(x)\,\mathrm{d}x$. Take an affine coordinate transformation $y=ax+b$. 
Consider the third raw moment in $y$-coordinates and pass to $x$-coordinates:
\begin{align}
m_3&=\int(ax+b)^3\,\mathrm{p}(y)\,\mathrm{d}y\\
&= a^3\int x^3\,\mathrm{p}(x)\,\mathrm{d}x
+3a^2b\int x^2\,\mathrm{p}(x)\,\mathrm{d}x
+3ab^2\int x\,\mathrm{p}(x)\,\mathrm{d}x
+b^3\int\mathrm{p}(x)\,\mathrm{d}x
\\
&= a^3 M_3
+3a^2b M_2
+3ab^2 M_1
+b^3
\end{align}
Imagine to write similar equations for $m_2$ and $m_1$. You get a system of three equations giving you $\{m_1, m_2,m_3\}$ as polynomials of $\{M_1,M_2,M_3\}$, with the coefficients involving $a$ and $b$. If you fix all six moments, you may be able to choose the coefficients so that two equations are satisfied, but in general the remaining one won't be. This means that you need a transformation involving more than two coefficients. (You may still not be able to find a solution though; this is related to the moment problem.)

Otherwise, an everywhere-positive density function can be transformed into any other everywhere-positive density function by an appropriate coordinate change, so in principle you can change the moments in any way you please (within limits related to the moment problem).

Regarding the other part of your question, I don't know of any canonical transformations for skewness and kurtosis, but maybe there are.
This is a tricky and fascinating topic. The fact is this: from "the point of view of the distribution", which is a measure, the manifold upon which it's defined doesn't need to have any additional structure (just a measurable space). For us to speak about a mean, the space needs to have an additional convex structure (which locally implies an affine one). With such a convex structure we can't speak about a second moment or variance. To speak about a variance, the space needs to have a quadratic form defined on it (quadratic forms can be defined on a convex space, even if it isn't a vector space; they have just slightly different properties from forms on a vector space). And so on. So usually the space has some additional structure that makes sense in the specific problem. The transformations we consider have to somehow be compatible with that structure to make sense.
This is also why we standardize the first and second moments of a normal distribution, but don't standardize higher moments: we could, with an appropriate transformation, but the distribution would not be a normal distribution anymore (that is, one belonging to the normal family). And how can we say that a distribution is a "normal"? We need (1) an affine structure and (2) a quadratic form on that space. [I hope this makes sense to you, sorry maybe I'm being too concise.]
