I am a native American English speaker. To improve my SAT verbal score, I studied Latin. Is there an analogous ancient language for Russian?
Old Church Slavonic is such a language. It is the earliest Slavic language of which we have written records, beginning with the end of the 9th century AD, and Old Church Slavonic (OCS) is so old that it is very close to the late Proto-Slavic language, so it displays the state of the language from which all the modern Slavic languages descend, not only Russian. Studying OCS greatly improves understanding of the phonology, grammar, and lexicon of the modern Slavic languages. Most texts in OCS are religious, though.
Also, OSC had a tremendous impact on the Russian language. Lexical (words) and morphological (prefixes, suffixes) borrowings from OCS are abundant in Russian to such an extent that their number partially obscures the fact that Russian is an East Slavic language, making Russian look much like a South Slavic language, since OCS is South Slavic. Studying OCS will improve your understanding of the Russian vocabulary and its etymology. Naturally, the OCS grammar is more complicated than that of Russian, in a similar way as the Latin grammar is more complicated than the Spanish grammar.
Here is a good book for a start, Old Church Slavonic Grammar, 2001, by Horace G. Lunt.
In Russian Orthodox church, OCS in its modern, phonetically and morphologically Russified version is still used, it's called церковнославянский язык, Church Slavonic.