1) You should change every diaper you can from as soon as you can. It's called division of labor. Especially if your womenfolk (or in the case of non-polygamists, womanfolk) are breastfeeding every couple of hours, they may feel somewhat tied down, put upon, exhausted. (You'll know this is the case because they'll tell you about it. ALL THE TIME.) Taking over diaper duty gives you some real skin in the game.
2) As for what diapers to buy and where, I have two words for you: Cost co. Two more words for city-dwelling dads: suburban relatives. They're the ones with Costco cards who you're going to press into service to get you the biggest damn boxes of diapers you've ever seen. We've found that the kid begins to outgrow a diaper size just as a case runs out.
Stay tuned for a separate diaper test drive post, where I'll try to decipher the rather arbitrary branding conventions (Swaddler vs. Baby-dry? What's the deal with that?). In the mean time if anyone has a clue, post away.