You don't need to "learn the languages" per se. That is, you do not need to learn how to create utterances in Ancient Greek or in Latin as that would call for learning all the declensions in those languages which are not relevant to the semantic impact they have on/in English on nouns or verbs.
What you do need to learn are the suffixes and prefixes coming from both and many roots of words to quickly help you with vocabulary in English.
All these examples below come from Reading Rockets , which has a pretty comprehensive view of all these roots, suffixes and prefixes encountered in English.
Many English words are formed by taking basic words and adding
combinations of prefixes and suffixes to them. A basic word to which
affixes (prefixes and suffixes) are added is called a root word
because it forms the basis of a new word. The root word is also a word
in its own right. For example, the word lovely consists of the word
love and the suffix -ly.
In contrast, a root is the basis of a new word, but it does not
typically form a stand-alone word on its own. For example, the word
reject is made up of the prefix re- and the Latin root ject, which is
not a stand-alone word.
For example, see the chart:
Common Latin Roots
Latin Root Definition Examples
ambi both ambiguous, ambidextrous
aqua water aquarium, aquamarine
aud to hear audience, audition
bene good benefactor, benevolent
cent one hundred century, percent
circum around circumference, circumstance
Common Greek Roots
Greek Root Definition Examples
anthropo man; human; humanity anthropologist, philanthropy
auto self autobiography, automobile
bio life biology, biography
chron time chronological, chronic
dyna power dynamic, dynamite
Some prefixes are from Latin and others from Greek:
Prefix Meaning Origin English Examples
a-, an- not, without, (having) no GREEK, anemia, atheist, atypical
ab-, a-, abs- away from LATIN, abnormal, absent, abstain, aversion
aer- air, atmosphere GREEK, aeronautics, aerosol
ambi- both, on both sides of LATIN, ambivalent, ambidextrous
ant-, anti- against, opposed to, preventive GREEK antagonist, antibiotic, antonym
audi- hearing, listening, sound LATIN, audible, auditorium, auditory
The full list can be consulted here: Greek and Latin Prefixes to word_Excel Institute
Please note: Ancient or Modern Greek is irrelevant as most of the terms are the same in both. One example is the word for woman: γυνή, which is the basis for the word gynecology in English. And -λογία (-logía) is where logy comes from as in biology, philology etc. etc. etc. [Wikipedia]
If you click through the links provided in this answer, you will see the suffixes, prefixes and roots from Greek and Latin you need to help improve your vocabulary in English.