The roots of English words, as we all know, mainly come from Greek and Latin. There is no doubt that knowing the Latin or Greek roots of English words greatly helps in memorizing them. For instance, "poly" means "many" in Greek, which makes remembering the meaning of words like "polyglot" or "polygraph" much easier. As it happens, both German and English are deeply influenced by Latin and Greek culture, so I want to know know whether they share some of the same word roots, and if yes, how many; will this make it easier to learn German?
English and German are Germanic languages. This means that their grammar and core vocabulary originate from Proto-Germanic, which is also at the root of Swedish, Norwegian, Danish, Icelandic, Dutch...
Latin and Greek are not directly related to English or German, although Latin, Greek, English and German are all Indo-European languages.
Words with Latin or Greek origins are many in English (there are especially many words that come from French, because French was the language of the elites in England for some centuries), but English does not come from Latin.
There are words of Latin and Greek origin in German, too, but not nearly as many as in English. You will often notice there are multiple English words referring to the same concept, at least one from French, and another from Germanic (they won't necessarily have the same meaning, and they will often be used in different registers, the French root usually belonging to a higher register). In many of these cases, German only has the corresponding Germanic word. Nevertheless, since both English and German are Germanic, knowing English will help you make sense of a few German words: these will often be more basic, core words than the ones that come from Latin or Greek, so they may be more useful in everyday speech.
If you're counting by pure number of words, it might be true that English has more roots from Latin or Greek than from any other languages, just because of the sheer number of scientific coinages—e.g., the names of the vast majority of biological taxa, and even many of the equivalent common names.1 Most of those words are shared across many languages, including German. So, a dictionary of every word ever used in English and an equivalent one for German might actually have a lot of common Latin- and Greek-derived entries.
But if you're counting by usage, it's very different. How often do you use the word "protostome"? Even if you're a zoologist, probably nowhere near as often as the word "and", which is a native word inherited from English's ancestors all the way back to proto-Germanic. Or "the", which is an Old English coinage, but derived from a root ("se") that again goes back to proto-Germanic. The same is true for many everyday words: "everyday" is a Middle English coinage from "every" and "day", which both go back to proto-Germanic roots, as does "word". Your own examples, "polyglot" and "polygraph", may be more common than "protostome", but nowhere near as common as "everyday" and "word".
In between these two extremes of "panarthropod" and "the", English does have a lot of borrowings. And, while many of those borrowings are from Old Norse (like "root"),2 which ultimately goes back to the same proto-Germanic as English, many are from Norman French and Middle French, like "native" and "extreme", which ultimately go back to Latin.
But these are much less common in German (because Germany didn't have a Norman Conquest). So, knowing those words—or knowing their Latin roots—will rarely help you with German. In fact, most French-derived English words sit alongside Germanic synonyms, or at least similar words—and it's those Germanic words that will help you understand German, while the French words just get in the way. For example, knowing "cow" is going to help you a lot more with German "Kuh" than knowing "beef", or the ultimate Latin root "bos".3
There are a few exceptions—e.g., English "state" and German "Staat" are both ultimately borrowed from French. But even there, what you want to know is the English word "state", or the French word "etat", or the Old French word "estat" that all three ultimately come from. Going back even farther than that, to the Latin root "status", makes it less useful, not more.
If you're reading a lot of scientific papers in German, knowing the Latin and Greek roots will help a lot—and, best of all, it will especially help with the hard words. But if you're reading newspapers, personal email, etc.? Not so much.
1. Consider "Hypsibius dujardini", "Eutardigrada", "true tardigrade" or "water bear", "Tardigrada", "tardigrade", "Tactopoda", "tactopod", "Panarthropoda", "panarthropod", "Ecdysozoa", "ecdysist", "Protostomia", "protostome", "Nephrozoa", "nephrozoan", "Bilateria", "bilateral", "Eumetazoa", "true animal", "Animalia", "animal"—every root in that chain of clades except "true", "water", and "bear" comes from Latin or Greek.
2. Also some from Dutch, another Germanic language, especially for nautical terms.
3. Unless you're able to reconstruct a common root between proto-Indo-European "*gʷṓws" and "*h₂uksḗn"… but if you can do that in your head just from hearing the words "cow" and "beef", you really shouldn't need any help understanding German…
As you are interested in seeing the relation more clearly between English and German while learning German: I ULTRA recommend the German Language courses by Michel Thomas.
He explains the similarities through the different way the languages evolved, with a very little limited set of phonetical variation. The similarity between English and German is clearer.
e.g. with the variation German <-> English:
- t <-> d
- g <-> y
we can clearly see :
- tag == day
- sag == say
- tun == do
Here the link at the beginning of his course where he concentrates on the similarities, and namely the shift of consonants which I was refering to:
- group: d - t - th - s - z
- group: b - p - ph - f - v - w
- group: g - gh - c - ch - k - ck - y - g
It made it much easier for me to learn or recognize german words associated to their English counterpart. I had found this really helpful.
Otherwise, as a foreign English speaker, I refer to this simple rule (which, like all exaggeratedly simple rules, is not entirely true).
- English words of 1 or 2 syllables come from Anglo-Saxon languages
- English words of 3 or more syllables come from French
Once speaking about this rule, I have been indicated that the Roman invasion had also left its heritage of Latin words into English also. e.g . these are words related to battle, weapons and fight, if I remember correctly.
The influence of French comes both from the Normans invasion, that ruled over England, which gave English words associated to nobless and feudalism. (It's one of the reason of having "pork" as cooked "pig meat", as Norman lords were served by english servants). And later because French was the language of the European Monarchies: e.g. much words dealing about the way to lead a state comes from French influence of these times.