So, I followed Stephen C. Carlson's advice and checked out TextKit. I was planning on doing that anyway, but knowing that whatever they had was OK was a good thing and it prompted me to do it sooner rather than later.
I'm poking through the first bits of Latin for Beginners by Benjamin L. D'Ooge, at least for now.
I'm realizing is that there are a lot of rules having to do with pronunciation. Penults, antepenults, dipthongs, short vowels, long vowels, vowel 'quantity' and syllable 'quantity', etc. Then I realized: Greek has a lot of this same stuff, and I don't consciously think of it when I'm reading Greek ... I just read it. So that's encouraging.
One disadvantage is that I don't get to actually hear anyone pronounce Latin. So I poked around the internet a bit. I found GreekLatinAudio.com, which has some MP3 files of various books of the NT, read from the Greek NT and the Vulgate NT. But the volume was very low and almost inaudible, and the enunciation didn't seem to be that great (but the sound was pretty faint ... ). But it's better than nothin'. Maybe I'll burn some tracks on a CD so I can crank it on my stereo. Then the neighbors will really think I'm weird.
Question: Are there pronunciation debates amongst Latinists similar to the pronunciation debates one finds regarding NT Greek? (e.g. Erasmian vs. Modern vs. ... )
Update: Regarding the Latin at GreekLatinAudio.com, Stephen C. Carlson (Hypotyposeis) writes in the comments:
I just listened to the first part of Mark 1 at GreekLatinAudio.com. It does not conform to any of the three systems I outlined above, and the speaker's knowledge of Spanish (as a second language?) is constantly interfering with his pronunciation. I don't recommend this AT ALL.
He also provided a helpful synopsis of Latin pronunciation — more than I'd anticipated. Thanks, Stephen, for the helpful comments.