First, I should say I am only planning on CDing, I have no experience actually doing it yet, but I've really researched the couple of things you've brought up.
I have thirsties duo size one covers and flip covers. Both I was graciously given (most barely used!) but after all my research I think those are the ones I would have picked anyway. I did buy a couple econobum covers before I was given the other ones, and I'm not really impressed with them, they seem very flimsy. I was also given a bunch of stay-dry flip inserts but I am also planning on buying/making some prefolds and fitteds for better coverage. You can absolutely use whatever you want inside of a cover, that's the beauty of it! I have a few of these prefolds in the works, by Adventures in Fluff, if you're interested in making some. There's also a handmade baby items thread way down on the first (or on the top of the second) page where some of the other women have discussed making diapers.
My question for all you seasoned veterans is about detergent. What have you had success with? I will want something natural and Cruelty-Free.
Oh, and one more! For those who brought their CDs to the hospital or started as soon as you got home, did you have any problems with the umbilical cord? We are debating how long we want to use disposables for before switching. I don't have any newborn CDs and I don't want to bother the umbilical cord by using the thirsties cover.