So, I have two related questions for you. I think no previous knowledge of linguistics is necessary for the discussion.

1. Does the knowledge of regular sound changes help with learning a foreign language related to a language you already know?
For instance, let's say you know English very well. Does it help you with learning German to know the following and similar rules (they apply to the Germanic words and very early Latin borrowings):
English "t" corresponds to German "z", which is read "ts", in the beginning of a word (two-zwei, ten-zehn, tooth-Zahn, tongue-Zunge, twig-Zweig…) and after "r" (heart-Herz…), but corresponds to "s" otherwise (water-Wasser, it-es, that-das, what-was…) except after "s", when it corresponds to "t" (star-Stern, stone-Stein…)
English "th" corresponds to German "d" (three-drei, that-das, thick-dick, mouth-Mund, death-Tod…)
English "v" corresponds to German "b" (seven-sieben, give-geben, live-leben, have-haben…)
English "d" corresponds to German "t" (desk-Tisch, word-Wort, god-Gott, ride-reiten…), but English "nd" corresponds to German "nd" (wind-Wind, hundred-hundert…)
English "oo" corresponds to German "u" (book-Buch, foot-Fuss, too-zu…)
English "ou" corresponds to German "au" (house-Haus, out-aus, show-schauen…)
English "ea" usually corresponds to German "o" (ear-ohr, east-Osten, easter-Obster, bread-Brot, bean-Bohne, death-Tod…)
I believe this was enough for anyone to get the basic idea.

2. Does knowing an archaic language from some family help with learning modern languages from that family?
For example, I've heard that, if you know Old Church Slavonic, you can basically understand all Slavic languages. Is that true?

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Offline Dither

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I can speak two languages, English and bad English.

But I do remember hearing that understanding Gramatical Gender can shortcut the learning process.

https://en.m.wikipedia.org/wiki/Grammatical_gender

If you have a mind for becoming a polyglot, and are cool with endless reps and carrying around flash cards, then there's never been a better time to jump on the language bandwagon, what with all the new and cheap learning methods and apps coming out (like memrise for example)

A lie will make it around the world before the truth has time to put on its shoes.

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Online Pete Svarrior

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2. Does knowing an archaic language from some family help with learning modern languages from that family?
For example, I've heard that, if you know Old Church Slavonic, you can basically understand all Slavic languages. Is that true?
It's kinda true, but knowing a modern Slavic language would yield much better results.
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Being an Indian, I can speak about 4 Languages - English, Hindi, Marathi & Marwadi

I am trying to learn German as of now.

In my experience, knowledge of linguistics surely helps in understanding the difference but it is only useful if one is conscious of it in daily practice.
I joined a coaching institute which actually helps me improve my pronunciation and not only do they teach the linguistic differences but also make me conscious of it.

For example - If you've studied syntax, you may find it easier to understand how sentences in a foreign languages are being formed. :)

devils advocate

I learnt French at school and was pretty fluent when I went to Paris a few years ago (many many years after school) I was amazed that I remembered it but was chatting to the locals like a native! I then worked in Germany for 3 years and became pretty fluent at German to the same extent. Problem was that I lost all my French, seriously, gone! I guess my brain can only hold one additional language so for me the only way to learn a language is to concentrate on that one.

I am not fluent in any language other than my primary, but I've played at learning enough other languages that I can say a formal study of linguistics is probably not going to be as useful as an informal immersion in the culture of the language you are studying.

Language is not just the structuring of words. In some cultures, phonetics are joined with tones, so that the same word can mean several things depending on how you "sing" it.
Also, some Asian languages seem to be based on a vastly different understanding of communication compared to European languages. Sentence structure is completely different. Gender concepts are foreign. Even the concept of an alphabet is different.

It helps to first understand how that culture thinks before trying to study how they convey those thoughts through their language.

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Offline Parsifal

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1. Does the knowledge of regular sound changes help with learning a foreign language related to a language you already know?

It can help, but it depends on how you use the knowledge and whether you're the kind of person that learns best through making logical inferences. I learned Dutch, a language closely related to English, while outside the countries where Dutch is spoken, in part by teaching myself some basic linguistics. Within 6 months of coming to Amsterdam, I have achieved the highest level of language certification offered by the Dutch government.

However, when I tried to explain some of my technique to other learners in the Dutch class I took after I got here, I mostly got blank stares and/or looks of admiration without real comprehension, leading me to believe that this approach isn't for everyone.

Also, I found it most useful in developing an intuition once I already had a basic grasp of the language. I can't imagine it helping very much for learning the basics.

2. Does knowing an archaic language from some family help with learning modern languages from that family?

I would echo Pete's comment that knowing a related modern language is easier. This is the case for a couple of reasons.

First, languages within an area are not totally isolated, they will influence each other and often share innovations that were established after their divergence. For example, consider that most languages in Western Europe today use simple case systems with articles and word order used to convey ancillary information about nouns, despite the fact that both Latin and Proto-Germanic had complex and meaningful case systems with no articles and flexible word order.

There are not very many cases of a language being totally isolated from its relatives, simply because the technology for large numbers of people to travel large distances did not exist until relatively recently. One unusual example would be Finnish and Hungarian, which are members of the Uralic language family that are divided by many countries which speak Indo-European languages. Finnish has been much more influenced by Germanic languages than by languages that it is related to over the past millennium or so, primarily due to its history of Swedish occupation. That said, I do not speak either Finnish or Hungarian, so I could not tell you how similar they actually are.

The point is that nearly all languages have been influenced by their relatives after divergence, and so their last common ancestor is likely to be less similar to them than modern neighbouring languages.

Even without grammatical innovations, there are words in modern languages which simply did not exist in their ancient counterparts. For example, Classical Latin had no word for "tomato", because the tomato was not brought to Europe until the 16th century. It also would have had no word for "television", because the television was not invented until the 20th century. The things that we need our language to refer to have changed over time, and all modern languages have adapted, but ancient languages are suited to the era in which they were spoken.

On top of all of that, it is simply easier to learn modern languages because a) they have more native speakers to learn from, b) there are more resources available to learn from, and c) there is a broader and more active corpus of media in those languages. It would be far easier to learn and compare French and Spanish to each other than to learn Latin and then either one of them.

Now, if you are interested (as I am) in ancient languages as an intellectual curiosity and a source of ancient literature, then by all means study them, but there are far better options out there if your goal is to effectively understand modern languages.
Many consider these extraneous classes an important part of a well-rounded classical education.