I think you should begin with Classical pronunciation, since it is more complex and shares many phonemes with classical Attic Greek. Ecclesiastical pronunciation, while beautiful, shows signs of degeneration and progression on a continuum towards modern standard Italian. Diphthongs adopt single vowel values, compound consonant sounds begin a single mixed sound, and the distinction between short and long vowels disappears -- all become long. It is easier to go from Classical pronunciation, therefore, to ecclesiastical, which is more or less the same Latin but pronounced in a modern Italian manner. Besides, it is good to learn about classical Roman culture, history, rhetoric, etc.
I used to be very strongly opposed to Classical pronunciation, but now I see how the linguistic reasoning behind it is insuperable. As for the declaration of Iulius Caesar (Yoo-lee-ooss Kahy-sarh, not Chay-sar) after the Battle of Zela, he wouldn't have sounded as silly as modern United-Statesians with Anglophone accents trying to pronounce the words, so I wouldn't worry much about that. Spoken Classical Latin is very beautiful and sounds much more Mediterranean and quite different than most imitations I have ever heard from heavily accent-burdened Anglophones. Same with vicissem, particularly the stress would be different, as would subtleties of the pronunciation of the consonants.