3g makes no difference in connecting to your home router. I'm 100 miles from the nearest 3g tower, and I can connect to my home router, and consequently my Edge service over cell, just fine.
What type of wireless encryption does your home router have?
I will say, the first Captivate I had was guilty of the same problem. It was constantly dropping and throwing errors on my home wifi, among other things. Captivate #2 works flawlessly.
I also went from an iPhone 3g. I was extremely close to the iPhone 4, but after giving the Captivate a second shot, Apple can kiss my shiny red apphole.
Another quick note- If you aren't at risk of international roaming, such as near the Canadian border, turn data roaming ON. On the iPhone, it only prevents international data roaming. With the Android(wife's Backflip also does this), it will prevent roaming on partner towers. There is no additional charge for data on partner towers, and my iPhone didn't even distinguish them from AT&T towers. Doing this solved a major headache with my Edge data service.