What techniques do I need to use to solve solve the differential equation $\frac{dx}{dt}=v(x)$, with $x(0)=0$? And would this classify as an ODE or PDE?
For a couple of particular cases, we could use $v(x)=\exp(-x)$, $v(x) = \tanh(x)$, or $v(x)=v_0 + k H(x-X)$, where $H$ is the Heaviside function.
Ask yourself: do you see partial derivatives in the equation? If yes, it's a PDE. If not, it's a ODE.
As for solving, rewrite it as $$\frac{{\rm d}x}{v(x)} = {\rm d}t,$$so that $$t = \int \frac{{\rm d}x}{v(x)} + c,$$where $c$ will be chosen as to make $x(0)=0$. I don't think you can do better than this in full generality, but the algorithm goes: solve the integral on the right, and then algebraically solve for $x$, if possible.