My family just recently purchased a Kindle. Both my wife and I have separate Amazon accounts. Can we connect both Amazon accounts to the same Kindle so that I could make whatever purchase I want from my account and she could make whatever purchase she wants from her account, but have them go to the 1 Kindle?
No, a Kindle can only be associated with a single Amazon account. What you can do is download ebooks bought from your wife's account, then use the Calibre (free) application with the DeDRM plugin and remove the DRM from the book you bought with that account so you can install it on you Kindle (connected to your account), but it's not as convenient as having them downloaded automatically to your Kindle.