The Absolute Minimum to Remember: There are three different sets of rules. They mostly affect C, AE, and I. CH is pronounced like English K.
Most people don't speak Latin these days except to import a word or phrase into another language. So in some sense it doesn't make much difference how you pronounce it. Still, various college courses keep getting mixed up with Latin, and it may be comforting to know what is going on with it. Learn this once, and you can impress people (or at least yourself) for the rest of your life.
In Roman times there were, of course, dialects, foreign accents, and so on just as there are in modern languages. And when Latin borrowed words from other languages (usually Greek) Romans sometimes had differing conventions for spelling and pronouncing them.
In general three standards of pronunciation are of concern to us today.
Latin in English, i.e. Latin words and phrases (like "et cetera") used in English (with slight variations between British and American usage)
Church Latin, whether used in a Latin mass, or in choral music, or as phrases borrowed from these sources into English (like "Te Deum")
School Latin, i.e., reconstructed Roman Latin (taught in college Latin courses and probably close to what was spoken on a good day by an elite Roman afflicted by a slight speech impediment)
The Evolution of Ancient Latin
Early Latin made a distinction between long and short vowels, which is shown today in most Latin textbooks by marking the long vowels (cēna = dinner, vs. cēnā = at dinner). By classical times the vowel-length distinction was being lost. As the empire assimilated more and more speakers for whom Latin was a second language, distinctions of vowel length were lost entirely in most areas.
The loss of the long/short vowel distinction resulted in some formerly different word endings coming to sound alike, which forced writers to depend more and more on prepositions and word order to make their meaning clear. This process is a central feature of the evolution of Latin into the modern Romance languages (French, Spanish, Romanian, Italian, &c.)
Most Latin words you see outside of textbooks will not have vowel length marked. Since stress was partly a function of vowel length, some church texts mark stress in ambiguous cases with an arbitrary diacritic to facilitate use in liturgies (céterus or cêterus rather than just ceterus) and an occasional book will mark a vowel as long if it makes a difference to the meaning (cenâ = at dinner).
Since stress is usually not marked, however, you may hear Latin words stressed differently by different speakers. For example, the emperor Septimius Severus may have his second name pronounced Séverus by one expert and Sevérus by another. (The second is technically correct.)
So here are some "rules" that generally won't lead you too far astray.
If there are two syllables, stress the first one (céna).
If there are more syllables, the one second from the last —called the "penult," as in "penultimate"— gets the stress if has a long vowel, and the one before it (the "antepenult") gets the stress if the penult has a short vowel (Septímĭus Sevḗrus). Since you probably won't know whether the penult has a long or short vowel, you have to take your chances (Sevḗrus or Sévĕrus).
The coward's way out is to speak very softly and hope a lot. Another approach is to do whatever sounds best in English. After all, that's the language you are speaking!
A = generally like A in "father."
E = anywhere between A in "fate" and E in "fed," your choice.
I = usually rhymes with "eye" in words borrowed into English (alumnus, alumni; Paranthropus boisei), but properly rhymes with "see" or resembles the I in "it" in any other context, such a full Latin sentence. ("Homo sum; humani nihil a me alienum puto." [I am a human; nothing human is alien to me.] —Terence)
O = generally like our long O, as in "goat."
U = generally like our long OO, as in "boot." The Romans wrote this with the letter V, which they also used for the consonant V (which they pronounced like our W).
Y = often pronounced to rhyme with "eye" in words borrowed into English, but rhymes with "see" otherwise. (The Romans, like us, used this letter to transcribe a Greek sound that we otherwise often transcribe with Ü.)
AE = pronounced to rhyme with "see" in borrowed words in English; the Romans pronounced it to rhyme with English "sky." In Church Latin it rhymes with English "say."
Caution: AE in ancient Latin rhymed with English "sky." But in English we often use that vowel to render Latin I. This can result in confusion. A female alumnus is an alumna, and the plural is alumnae. Romans would have pronounced that AE to rhyme with "sky"; in English it comes out rhyming with "see." Ancient Romans, hearing the latter pronunciation, would have assumed the spelling was "alumni" and that the individuals were male. (Whoever said language was such a great way to communicate?!)
AU = pronounced to rhyme with "plow."
EI = rhymes with "bay."
EU = really does sound like "eh" as in "heh heh heh" plus "ew" as in "phew." It's ugly, but that's just how it is. (Think of it as a Eu-ropean problem.)
Most consonants can be pronounced like their English equivalents. Here a couple of marginally trickier ones.
C before E, I, Y = Like S in words borrowed into English. Like K for the ancient Romans and in connected passages of Latin.
Caution: The old Roman K sound for the letter C is coming into use for more and more Latin words used in English. (Some people speak of the Celts as "selts" but most people these days call them "kelts.") In Church Latin, which follows Italian, this is pronounced like an English CH, and this comes into English in borrowings from Church Latin (like "Ecce Homo," where the CC is pronounced like English CH). Similarly, the spelling SC pronounced SK (Roman), SS (English), or SH (Church).
C before A, O, U = Like K.
CH = like K.
G before E, I, Y = Soft G as in "Gerald" in English borrowed words and Church Latin, but hard as in "garden" for the Romans.
G before A, O, U = Hard G, as in "garden."
QU = like KW regardless of what comes next (unlike French or Spanish).
J = like English Y (but more often written I).
W = like English V (but more often written V).
X = like English X (but occasionally pronounced K in clearly Greek borrowings).
PH and TH = as in English. These were used by Romans to represent exotic Greek sounds (shared with English), but Latin speakers in some areas had trouble with them and they tended to devolve into a simple P or T. Standard school Latin, like Church Latin, usually pronounces them F and T.