"I know nothing about women."
Translation:Non so niente di donne.
"delle" is kind of a contraction of "of + the" so it adds the definite article, which is acceptable. I think it does, however, change the meaning slightly. "I know nothing about THE women" or "of the women" might be taken to mean a specific group of women rather than women in general. That is my understanding, but full disclaimer, I'm a beginner, as well.
Four years late, so I'm guessing you've figured this out by now, but I believe that "circa" and "quasi" mean "about" in the sense of "approximately": "It's about a five mile walk". (Circa actually has this same meaning in English, as well, as does quasi- as a prefix.)
So while those are translations for "about", they're not quite right in this case where "about" means "relating to".
This is a difference that many European languages have, including German (kennen vs können) and Dutch, French (savoir vs connaître) and also Italian. Conosco mostly means "to be acquainted with, to have met" and in general is used for people. You use it for "know" in the sense "do you know that girl over there?" or "I don't know you". Also see this discussion on Duo: https://www.duolingo.com/comment/2088826
It's similar to Spanish. "Conoscere" is for intimate knowledge, "sapere" is for facts and figures.
If you know someone, it's "conoscere". If you know about someone, it's "sapere".
First, "conostro" is not a word in Italian. You're thinking of "conosco".
Second, that's the wrong verb. You want "so".
Third, you're missing half of the negation. It needs to be "non so niente di donne". It's called negative concord and it's somewhat similar to adjective agreement.