Just want to point out that NYU and Columbia are not part of the SUNY system.
Generally speaking, if your CC is accredited, it is highly likely that your credits will transfer. However, HOW they will transfer (for general ed or for specific major requirements) is more up in the air. Some of the SUNYs have “transfer” converters similar to what California does - look under the Transfer Student Admissions tab on the school you are interested in transferring to.
You are correct that generally you will need a C (sometimes a C- is okay, but you should aim AT LEAST for a C). Obviously the higher your grades at the CC, the more likely you will be able to transfer to your desired school (and NYU and Columbia will required a very high GPA).