The above write-up is correct as a simple guide: what I write here amplifies and justifies it.
Evidence for pronunciation comes from alternations in inscriptions, in explicit statements by contemporary grammarians, in phonetic changes in certain environments in a word, in transcriptions into other languages, and much more. Some people say we "don't know" how an ancient language sounded, as if only a tape recording would suffice. But within an abundantly recorded language like Latin, we do know with a high degree of certainty, and changes are often datable.
The short vowel A would have been the same as long Â as in father, but shorter: the sound in fun for most speakers in southern England and Australasia.
C was always pronounced K in the classical period. The first evidence for the shift to the later TCH and TS and S sounds appears in the fourth century. Sometimes K was used in inscriptions, e.g. pake for pace. Greek transliterations always used kappa, e.g. kênsôr and Kikerôn for censor and Cicero.
Likewise G was always the hard sound of get up until then (c. 500 CE), when the softer gent sound appeared.
However, in the group GN it appears to have been NGN as in hangnail (and the alternative spelling NGN was sometimes seen): so magnum 'big' was not as it's presently said in English. This seems to have been 'corrected' to G + N at a late stage, and in the name Gnaeus it was possibly silent, at least in common speech. The group GN didn't have the modern Italian or French NY sound.
The change of V (consonantal U) from a W sound to a modern V sound was much earlier; and although it was a W in Cicero's day, it evidently shifted to V about a century later. Silver Age writers like Tacitus and Pliny probably used a Spanish V sound, bilabial, not quite the English V. Inscriptions sometimes confused it with B at this time.
R was rolled. Grammarians called it the littera canina 'dog letter'. In the Old Latin period it might have had a different pronunciation in some places, because some instances of R came from an earlier S, e.g. intervocalic honôsis became honôris. In Greek, initial and doubled R were voiceless: Latin represented this in borrowed words as RH and RRH but presumably only people who knew Greek actually said them in the Greek way.
CH, PH, and TH occurred mainly in Greek words, representing the letters chi, phi, and theta. This implies the Latin letters C, P, and T were unaspirated as indeed they are in modern French, Italian, and Spanish, whereas CH PH TH were like the English sounds in car par tar. But they occurred in a few native words too, like pulchra 'pretty', lachrima 'tear', Gracchus, triumphus. A few such might be explainable as borrowings from Etruscan, which also had aspirates, but lachrima, also spelt lacrima and lachryma, is related to Greek dakru with unaspirated K, so it was a development within native Latin words. Cicero discusses it and actually changed his mind about which words to use aspirates in and which to leave plain; one of Catullus's poems mocks a speaker for their Greekified use of CH. Both writers make clear that the use of aspirates was an affectation.
M was normal in most positions, but at the end of a word it was not a consonant; rather it indicated that the preceding vowel was a long nasal vowel. It had started as a consonant M here, but had been lost in Old Latin, well before the classical period. For example, a third century BCE inscription writes oino and optumo for classical unum and optimum. After the classical period the nasalization was lost entirely, so unum became Vulgar Latin uno The vocalic quality is also shown by e.g. disputandumst in Plautus, for disputandum est.
The same was true of N before F and S, so infans 'child' was pronounced îfâs with two long nazalized vowels, and consul was côsul. This loss of N also happened early in Old Latin, and the nasalization disappeared entirely in Vulgar Latin. However, there seems to have been variation here, and some speakers used or reintroduced a consonantal N: perhaps educated speakers influenced by the spelling.
AE was originally a diphthong AI, and shifted to AE (ending in an E-like sound) in the second century BCE. (The difference is slight: both are close enough to the sound of English aisle.) It went on to become a long Ê, and the diphthong AU became a long Ô, apparently in or a little before the classical period, but this was long regarded as rustic, and grammarians tried to enforce the diphthongal pronunciation.
The consonant I (= J) was the Y sound of yes, you, never an English (or French) J. That change occurred only after Vulgar Latin began splitting into different Romance languages. Between vowels it was a double consonant: so eius, cuius, maior were pronounced eyyus, cuyyus, mayyor. The spellings maiior etc were sometimes seen.
Y and Z were borrowings from Greek, representing sounds that didn't occur in native Latin. At first words containing them were borrowed with the closest Latin sounds, U and S, but as they became more sophisticated, they used the authentic Greek sounds. Later, after the classical period, Greek Y changed to an I, and this is reflected in later Latin borrowings. Z (zeta) was a ZD sound in classical Greek, and later changed to the English Z sound. Between vowels it probably underwent an intermediate ZZ stage, because it makes the preceding syllable long in Latin: as in Mezentius, which was pronounced Mezzentius. (But see zeta for some problems with this analysis.)
The definite reference for Latin pronunciation is:
W. Sidney Allen, 1965, Vox Latina, Cambridge.
My thanks to Gone Jackal, our real expert on Latin, for offering evidence and suggestions on several points, and my apologies for not incorporating them sooner.