It depends on where you were taught/who taught you. There’s no universal “Latin” course. However, if you are being taught Latin right now, there are a few ways to tell by pronunciation whether you’re learning Classical or Medieval Latin.

In Classical Latin, I’ve been taught that pretty much every consonant is pronounced as if it were alone in English. The only changes are that a “v” makes a “w” sound, an “i” can act as a consonant before a vowel, and a “c” is always pronounced with a hard “k”. While there are some other rules with poetic elisions and whether -io is one or two syllables based on meter, it is straightforward consonant wise.

Vowels are a bit more different from English. Ideally, “a” is always pronounced as “ah”, “e” as “ey”, “i” as “ee”, “o” as “oh”, and “u” is “oo” if not a “w”. Additionally for diphthongs, the only not so straightforward ones I remember are that “ae” is “aye”, and “au” is “ow”.

Let’s put this in practice so you can hear yourself: “Ave, Caesar! Morituri te salutant.” (Hail, Caesar! We who are about to die salute you; gladiators would say this before fighting.) Read in Classical Latin, it would sound like “ah-way, kaisar! moreetooree tay sahlootahnt.”

On to medieval/ecclesiastical Latin, the only other Latin usually taught. I’m only drawing from a brief unit on medieval pronunciations, and I will disclaim that the changes from Classical Latin did not occur at one time or every place, so perhaps slightly different pronunciations are taught.

Medieval Latin is a lot more complex, with lots of new sounds. A “ti” goes from Classical “tea” to Medieval “tsi/zi”. A “cc” goes from a Classical “k” to a Medieval “ch”. C’s can become soft with “s” sounds like in English. An “ae” is now an “ee”. “I” can became a fancy new letter “j”, with its standard English pronunciation. Vowels are a little less strict. This list is certainly not comprehensive, but hopefully you can spot differences.

Now let me make something church-y up so you can read aloud: “Ave et ecce.Jesse, qui filius optimī piae matris Dominīque est.” (Hail and look, Jesus, who is the son of the greatest Lord and holy mother. After weeks of Ovid, I couldn’t resist that chiasmus.) Aloud, this would sound like “Ahvey et echay, Jesse, qwee filius optimi pieh matris Dominiqway est.”

This is neither the most comprehensive nor the most sophisticated explanation of Latin linguistics, but hopefully it both answers the question of yes, Classical Latin that Caesar and Cicero spoke is taught today, and also how to tell the difference if you’re in a Latin course right now. Vale!

Related follow up question: Was Classical Latin actually spoken as a home language, or was it simply the equivalent of a 'proper accent' that many people do today for radio, TV, movies, etc. etc.?

I am aware the Vulgar Latin spoken at the time of Caesar was already quite different than the Classical form, and so I must ask whether he might address his soldiers in their vulgar form, or whether he'd have been born and raised entirely in the Classical pronunciation habits?