Is my proof of straight line being the shortest route from one point to another correct? Every proof seems to go above my head as I'm not thorough with calculus or what is being talked about in this similar question in which the author proves it using complicated terms. As a result I tried proving it on my own. 
I think it has to do with triangle inequality. Can we say that any curve joint to the two points other than the straight line as sides of a polygon? Can we say that the sum of the sides of that polygon will always be more than the straight line because of triangle inequality theorem? If we can say that, it is proved much more simply than any other proof I have seen yet.

I think it works for polygonal paths, but this certainly does not show the result in full generality. Indeed, a different way to go, might be to define a path to be a continuous map $\lambda:[0,1] \to \mathbb R^n$. 
We need a meaningful notion of "length" in this context, and if we take $d(x,y)$ to be the usual euclidian distance one can propose the following definition

The length of a curve $\alpha:[0,1] \to \mathbb R^n$ is 
  $$\sup_{0=t_0<t_2<,\dots,<t_n=1}\sum_{i=0}^{n} d(\alpha(t_i),\alpha(t_{i+1}))$$

Indeed this makes the proof quite easy, and it is really the "limiting" analogue of your own proof:
A valid partition is given by $n=2$ where the partition is taken to be $\{t_0,t_1\}$ of the interval, in which case the definition of supremum implies that any other path is greater than the straight one.

The more familiar definition of arc length $\int_{[0,1]}d(x(t),y(t))dt$ is equivalent to the one given for a large family of curves (and in particular class $C^1$.) See Rudin's Principles, 6.27.
