Lagrange mechanics has the same form no matter what coordinates you use to describe your system. It uses so called "generalized" coordinate $q_i$ and their first derivatives. On the other hand, Newtonian mechanics deals mostly with vectors in 3D space and those are highly dependent on coordinates you choose to describe them.
It depends really what you want to know. There is a whole field of mathematics (called differential geometry) which intoduces Christoffel symbols and this abstract index notiation. One can use it to describe different views from different coordinate systems and transformations between them (that is what you see here). There is really a lot of mathematics you have to go through first in order to grasp what is going on here. I am still pretty green in the field of differential geometry, but I would say that if you are interested in Euler-Lagrange equation, you don't have to worry about Christoffel symbols.
I can recommend you a great introductory book for Classical mechanics from John R. Taylor where Lagrange and Hamilton formalisms are gently introduced and used to solve various relatively simple problems.
If you want to learn differential geometry, I am currently reading Sean M. Carroll's book on General relativity where differential geometry concepts are introduced and used.