I find this fascinating too. From where I live, I've noticed and started learning more about Sanskrit's influence on European and Slavic languages. Maybe if I studied Sanskrit, it could help me learn more languages quicker, with a somewhat common vocabulary of root words.
Sanskrit and most European languages do share a common ancestor. But Sanskrit is not that common ancestor, and the influence of Sanskrit itself on European languages is trivial, mostly limited to words relating to Indian religions, yoga, and so on.
I think this confusion arose because it was similarities between Greek, Latin and Sanskrit that first alerted scholars to the existence of a common ancestor.
Cool. You should go for it. Having studied Sanskrit earlier for some years, I can say that it is a fun and interesting language to learn, and also not very difficult, since the grammar is very regular and systematic.
(That's a scholarly article, don't go by it, though you could skim it, just giving it for background info. Try out the language instead.)
There are a handful of rules, and the rest of the stuff builds upon that, the way programs build upon the handful of syntax and semantic rules about statements in programming language.
An interesting point mentioned in the article is that "Word order is free" in Sanskrit.
While not a linguist myself, I've noticed that the meanings of sentences in Sanskrit seem to be intelligible and mean the same, even if you switch some of the words around. I don't know if that is true for any other languages.
I wonder if that point has any connection with the concept of context-free grammars.
