This is a Google Translation:
English: We can talk in Chinese, English, or Spanish
I eat wondering: Why isn't it necessary to add と after each list item?
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Attaching と to all alternatives except for the last one is the most common way when listing. Considering what you want to say in English
We can talk in Chinese, English, or Spanish, the direct translation is weird. An important thing to know is that the usage of comma in Japanese and English is different. Those links are interesting:
In Japanese, we would have
私達は中国語と英語とスペイン語が話せます to emphasize that the people in this group can speak all the three languages, therefore they can talk in any of the three languages (at least, this is what I understood from the statement in English). 話せます is fine once we want it to mean that those people can speak or can talk the said languages.
Also, I think 私達は中国語と英語とスペイン語を喋ります would be an alternative, but if you use が, then we would have 私達は中国語と英語とスペイン語が喋れます, but less formal than 話せます.
You could use 「が」 or 「を」in the first example, that is 「私達は中国語と英語とスペイン語が話せます」or「私達は中国語と英語とスペイン語を話せます」。どちらもほとんど同じ意味ですが、今では「私達は中国語と英語とスペイン語を話せます」を使っている人が多いと思います。
In conclusion, 私達は中国語と英語とスペイン語を話せます would be the best option for the translation. I don`t see a reason to use commas to separate the clauses, the commas can be completely left out, also considering that the clauses before are short. You could perhaps use commas in a similar situation to increase readability, but it is not the case here.
Concerning the particle 「も」, you can use it as well, but not simply replacing 「と」for「も」in the sentence 私達は中国語と英語とスペイン語を話せます. Using 「も」 to give the ideia of
also is valid, but 中国語も英語もスペイン語を話せます is incorrect, you should say 中国語も英語もスペイン語も話せます. If 「も」is better fit, I think it would depend on the context, that is not provided, but both cases can be used.