So how do we get "chonaic mé" out of the verb "feic"? I am trying to look it up in my grammar book. I see it is an irregular verb. It doesn't tell me why it is that way but I do see that you only use "chonaic" if you are saying you did see something if it is negative and you didn't see something you use "fhaca"?
Yes, "I saw" is chonaic mé and "I didn't see" is ní fhaca mé. And for the sake of completeness, the question "Did I see?" would be an bhfaca mé?
What's happening here is that a few of the irregular verbs have separate so-called dependent and independent forms. The independent form is used when the verb is used on its own (well, I guess you could say independently), and the dependent one when it follows most of the verbal particles like ní, an, nach, go, etc. After the relative particle a and after má ("if") you use the independent form, but otherwise it's pretty much everywhere the dependent one.
Actually I think you're already familiar with this phenomenon, so chances are that you can work it out by analogy when you're unsure which one to use: The dependent form of tá is fuil (and because most of the particles eclipse, you know it as bhfuil) and in the past tense you have independent bhí and dependent raibh.
So whereever you would say tá you use chonaic, and whereever you would say bhfuil you use bhfaca.