Any good dictionary will provide you a pronunciation guide. Online dictionaries have an advantage over print editions: they can embed audio files for additional help.
Pronunciation guides list all of these as valid pronunciations:
- /SHiˈkôgō, -ˈkägō/ (from ODO)
- ʃəˈkɑgoʊ ; tʃɪˈkɔgoʊ (from Collins, American edition)
- ʃɪˈkɑːɡəʊ (from Collins, British edition)
(If some those symbols are confusing to you, look here).
Anyhow, the "Ch" is almost invariably pronounced with an "Sh" sound, much like chagrin and champagne, although a few might use the traditional "Ch" sound, as at least one dictionary listed that as an alternate pronunciation.
As for the vowels, you may hear slight variations; regional accents may apply.
Probably the best place to answer a question like this one is Forvo, where you can hear numerous anglophones (in this case, 10) take a crack at letting you know how to pronounce the name of that Windy City.