It's not that light has any inertia, it's simply that when you view the path of something from a different frame, the path can look different. Light's having no mass does not affect this qualitative fact. The masslessness simply means that its speed is frame-invariant, not its direction of travel.
In particular, if for example, a light ray is traveling in the $y$-direction in one frame, then it will look as though its path is angled towards the negative $x$-axis in a frame boosted along the positive $x$ axis.
Mathematically, consider a light ray along the $y$-axis described by the following path:
\begin{align}
t(\lambda) &= \lambda, \qquad
x(\lambda) = 0, \qquad
y(\lambda) = c\lambda \qquad
z(\lambda) = 0
\end{align}
The Lorentz transformation tells us that in a frame boosted in the $x$-direction, we have the following relationships:
\begin{align}
t' &= \gamma\left(t-\frac{v}{c^2} x\right), \qquad
x' = \gamma(x-vt), \qquad
y'=y, \qquad
z'=z
\end{align}
so the path of the light ray in the new frame is
\begin{align}
t'(\lambda) = \gamma\lambda, \qquad
x'(\lambda) = -v\gamma\lambda, \qquad
y'(\lambda) = c\lambda, \qquad
z'(\lambda) = 0
\end{align}
Notice that the path of the ray has picked up a negative $x$ compoenent! This is purely because we are viewing the light ray in a different way. It's like if you were to view your computer monitor while sitting at your desk, it looks like its standing still, but you can make it look like its moving in any direction you please by yourself moving in an appropriate direction.
Notice, however, that in both frames, the path of the light ray satisfies the appropriate nullness condition;
\begin{align}
-c^2\dot t(\lambda)^2 +\dot{\mathbf x}(\lambda)^2 = 0
\end{align}
where dots denote derivatives with respect to the parameter $\lambda$ which indicates that the speed of the light ray is invariant.
When it comes to light any analogy with water is doomed to be completely wrong.
I word this so strongly because it is a common misconception that frustrated all of physics for several decades back around the turn of the 20th century. People kept trying to make analogies with water, and the result was always wrong.
What you are describing is what is known as the aether theory of light. People thought "well, light is a wave, and water waves need a medium to travel through, therefore there must be some sort of medium in which light travels." This is simply not true. The EM field is not some sort of physical substance that has a bulk velocity. When light is emitted, it is not perturbing some medium that is stationary in some reference frame. It propagates, and you can describe that propagation in any frame you like, but there is no "correct" or "natural" frame.
Michelson and Morley are most famous for designing the experiment that proves this beyond any reasonable doubt. Basically, they measured the speed of light in various directions, aligned with and perpendicular to the Earth's motion about the Sun. That speed did not change, not one bit. The Sun's rest frame is nothing special; the source may be moving with respect to it, but so what? Light moves away from its source, at the speed of light, in a "straight line," as seen by any inertial observer.
If you thought otherwise, you would have to believe that some inertial frames are more important than others; that some observers/sources are really, truly, absolutely in motion while others are truly at rest; that the shore of the pond was a more privileged thing in the universe than person running alongside it. Physicists eventually realized this was preposterous - the universe doesn't think the shore of the pond is more special than a person - and hence was born special relativity.
Best Answer
Everything moves in geodesics if not acted on by a force other than gravity -- this is an axiom of general relativity (the geodesic equation). Geodesics are straight lines in the absence of gravity -- this is part of the other axiom of general relativity (the Einstein-Hilbert action, or the EFE or whatever).
Light doesn't interact much with everything, except quite weakly with gravity, and with some miscellaneous scattering patterns, like those which allow you to actually see things, but those are quite pointy (reflection, refraction, etc. -- the paths are pointy as long as the scattering boundary is sharp), so you still see a bunch of straight lines.