I've been working through Athenaze at home, on my own, with no one around who knows a word of Greek. I think that Athenaze is meant to be used in a classroom, and that the teacher would fill in the blanks, especially in the readings. Here's what I've been doing.
I try to translate the readings as best I can, writing it all down, and feeling free to look up words I've forgotten, checking the summary grammer at the end of the book, and looking ahead in the chapter to see what new concepts are being introduced. I'm not sure that this is what the authors intended, but it works for me.
One day I felt really stuck, and not at all confident that I was "getting it," and I bought a copy of the teacher's guide for Athenaze. It tells a little about what each chapter is intended to teach the student, and has a translation, with some helpful notes, of each passage. So I translate a paragraph, and then compare my translation with theirs. Usually, it's not bad, which builds my confidence, and when I've missed the boat, I learn what I've done wrong - which is what's really important, I think.
I also picked up copies of the Oxford Pocket dictionary and grammars, and have sometimes looked something up there, but not as often as I thought I would.