The Absolute Minimum to Remember: Greek has no sound like English CH. CH is like English K.
Some 11 or 12 million people speak modern Greek (roughly twice the number who speak Norwegian or Mayan).
But normal people don't speak ancient Greek these days except to import a word or phrase into another language, such as English. That should mean that it doesn't make much difference how you pronounce it, but there are conventions, and you are better off following them, if for no other reason than because everybody else does. Learn this stuff once, and you can impress people (or at least yourself) for the rest of your life. (You can also avoid embarrassing gaffes.)
English transcriptions of ancient Greek words (or English words built on Greek roots —so-called "neo-Grecisms") tend to be based on the letters of written Greek, not the sounds of the language. That is convenient in certain ways, but some of the transcription conventions don't always use English letters the way they are used in other English words, and some Greek letters are not consistently differentiated in their English transcriptions. Here is a list of pointers. (Where Greek letters are given, they include both upper and lower case and are printed in blue.)
The single most important point to remember is that there is no CH (as in "cheese") in Greek. In transcribed Greek names, CH always comes out sounding like K. If you get that right, people will forgive most other missteps.
Greek words used in English, including proper names, are usually anglicized in speech. For example, the region spelled Boeotia comes out "bee-OSH-a" even though that doesn't sound very Greek. In general, if you remember that the spelling OE is pronounced like the EE in "see," you'll be fine (Oedipus = EE-duh-puss).
Like English (but unlike most languages), Greek had a TH sound. In Greek it was written with the letter theta (Θ θ). In English it is always spelled TH, just as you would imagine.
Greek had an F (Φ φ), but it is usually transcribed with the spelling PH, as in “philosophy.”
Some Greek consonants correspond to two-letter combinations in English. This is the case with the letter psi (Ψ ψ), which was pronounced PS, and with the letter xi (Ξ ξ), corresponding to our X.
However, in Greek psi and xi can occur as initial sounds of words, which would seem unpronounceable in English. Therefore in PS we write but do not pronounce the P when it is initial (as in “psychology”). Similarly when X (pronounced KS) is initial, we pronounce only the S part (as in "xenophobia" or "xylophone" or Xenophon).
The Greek letter that looks like X is named “chi” (Χ χ), which is pronounced “kai” (as in the “chi-square” test in statistics). The chi was a kind of very hard h-like sound. It is always transcribed with a CH in English. Greek did not have a sound corresponding to our CH, so whenever you see CH in a Greek word, pronounce it like K. (That is why “chaos” is pronounced “kay-oss” and not “chay-oss.”) Greek also had a real K: the letter kapa (Κ κ). In pronouncing transcriptions, it is pronounced identically with chi: both are pronounced K.
Greeks had two letters transcribed in English with the letter E. One was epsilon (Ε ε) which was pronounced like the E in “met”; the other was eta (Η η), which was pronounced like the EI in “freight.” Although specialists distinguish these by putting a long mark over the eta, most editors take it off again, so you are unlikely to be able to worry about the distinction even if you want to.
In English, however, a final E (alone or followed by S) on a Greek borrowing is usually pronounced like an English long-E: Andomache = and-DRAHM-muh-kee; Achilles = ah-KILL-eez.
Similarly, Greeks had two letters transcribed in English with the letter O. One was omicron (Ο ο), which was pronounced more or less like the AW in “law”; the other was omega (Ω ω), which was like the long O in “bone.” Once again, editors routinely suppress the long mark, so the distinction need not bother you.
The Greek letter upsilon (Υ υ) is usually transcribed with Y, but its original sound was like a French U or a German Ü. (Arrange your mouth as though you were about to say the U in "glue" and, without moving your lips, say the EE in "glee" and you will have it about right.) When anglicized, it is usually pronounced like an English U or like the word "you" by most English speakers.
Most of the diphthongs of ancient Greek are pretty self-evident, except that upsilon seems to lose its Ü-like quality in diphthongs and becomes merely a diphthong marker. Greek ου, is usually spelled OU in English to reflect its Greek spelling, but is pronounced like the OO in “cool.” (The island is spelled Sounion, but pronounced SOO-nee-on.) (Note: Greek had no letter corresponding to a regular U, but the sound was common, so the two-letter combination was common, and hence OU is common instead of U in our transcriptions.)
The Greek diphthongs ευ (epsilon-upsilon) and ηυ (eta-upsilon) are both transcribed as EU (as in “Europe” or “euphemism”), and both were apparently actually an E-sound followed by a W-sound in Greek (something like the sound we make when we are disgusted by something), but the EU spelling comes out in English speech like our pronoun “you.” That is why we say "Europe" the way we do. That is not very Greek, but it is conventional and it works okay.
Upsilon figures in a couple more Greek diphthongs where it is not transcribed and need not bother you.
Ancient Greek, as reprinted today, uses initial apostrophes —left-facing and right-facing— to show the presence or absence of an initial H, since there was no letter for it. (On computers and in many type fonts, the difference is extremely difficult to see.) In English transcriptions, when the H is there, it is transcribed H. When it isn't, it isn't marked. If you never take a course in Greek, you can ignore the matter. (If you do take a course in Greek, you will also learn lots of other little marks you can scatter over and around the letters. Until about 200 BC, the ancient Greeks used none of them. In fact they limited themselves to capital letters and usually didn't bother even to separate words.)
Many Greek words ended in -OS (-ος), but were borrowed into Latin with the ending -US and come into English wholly or partially in their Latin forms. Most people stick with the Latinized spellings. Take your choice.
Ancient Greek seems to have had distinctions of pitch —similar to Chinese tones— but apparently stress of the kind we use in English was unknown or unimportant. By about 200 BC the pitch distinctions were marked in some texts by means of diacritical marks that seem to have meant rising, falling, and rising-then-falling. Such markings disappeared by about AD 400, and there is no trace of "tones" in modern spoken Greek (although modern printings of ancient texts include the diacritics).
Most specialists believe that the appearance of the diacritical marks represented a refinement of the writing system to represent speech more accurately, while their disappearance corresponded with the replacement over time of pitch differences by differences in stress. For historians, the careful analysis of such details can be helpful in spotting dialect differences in inscriptions, sometimes attributable to population movements. (For an attempt to pronounce the first 52 lines of Homer's Iliad with reconstructed pitch click here.)
For Greek names used in English stress is not easily predicted from their English spellings (and does not necessarily correspond to where the Greek accent marks went). On short words, go for the penult (the syllable before the last): Achilles ( ͐Αχιλλευς) = ah-KILL-eez. On long words, go for the antepenult (the third one from the end): Telemachus (Τηλεμαχος) = tell-EMM-ah-kuss. You’ll be wrong sometimes, but not too far wrong most of the time.
Ancient Greeks referred to themselves as Hellenes ( ͑Ελληνες), i.e., descendents of an ancient king named Hellen ( ͑Ελλην ). Hellen married a prolific sea nymph and their many sons settled across Greece fathering various flavors of Greeks: Aeolians, Ionians, Achaeans, Dorians, Boeotians, and so forth. (Caution: Hellen, with two Ls, is not to be confused with Helen of Troy [ ͑Ελενη ], with one L.)
The Romans, from whom the English word "Greek" comes, called them Graeci or "people from Graea," the location of which is still not certain. Perhaps it is your destiny to discover this lost place. Here is your chance to win scholarly fame and to ride picturesque sailboats around the Mediterranean with National Geographic photographers while you do it.