(Expanding my comment into an answer)
When discussing phonology of Chinese relative to English (or some other language), there are a couple important things to keep in mind:
Sounds (which we try to represent unambiguously with IPA notation) are not the same as letters. The letter "t" in American English can correspond to several different sounds. For example, the "t"s in each of the following words would all be written differently in IPA: "tea", "steam", "beat", and "butter". Perhaps you would consider the "t" in "tea" to be the prototypical "t" sound. How is it written in IPA? Not as [t]! It's actually [tʰ]!
IPA makes a 3-way distinction regarding voicing and aspiration. Some stops are truly voiced (in the sense of having vocal chords vibrating before the stop is even released). For instance [d]. There are unvoiced-unaspirated stops, where the vibration-start and the release are more-or-less simultaneous (e.g., [t]). And there are unvoiced-aspirated stops, where a puff of air comes out after the stop has been released and before the vibration of the vocal chords starts (e.g., [tʰ]).
Chinese, English, and French all make different two way distinctions in voicing/aspiration.
- Chinese distinguishes [tʰ] and [t]. [d] is never produced but "sounds like" [t] to a Chinese listener.
- French (and Japanese, for that matter) distinguishes [t] and [d]. [tʰ] is never produced but "sounds like" [t] to a French listener.
- English distinguishes [tʰ] and [d]. [t] is never produced (in isolation at the front of a syllable, anyway), but (on its own), it "sounds like" [d]. However, for English speakers listening to someone who makes a [t/d] distinction (e.g., someone with a strong French accent), the [t] usually sounds like a [tʰ].
If you are unaware of the above, or you ignore it, it will be very confusing discussing the sounds in Chinese vs those in English. Here is my summary of how "z" and "c" sound, given the above:
Pinyin "z" is pronounced [ts] by Chinese speakers. If English is your native language, you are likely going to perceive it as sounding like [dz]. If you also pronounce it as [dz], that will probably "sound like" [ts] to Chinese listeners, so there won't be any miscommunication.
Pinyin "c" is pronounced [tsʰ] by Chinese speakers. If English is your native language, you will likely hear it as [tsʰ]. Note that the letters "ts" in English--for instance, in the word "tsar", assuming you attempt to pronounce the t in there--are pronounced [tsʰ]! "ts" is not pronounced [ts]!