There were problems Saturday as well, just because someone had no problems does not mean "all is well" for everyone.
Many, many original iPhone owners had bricks for several hours. A friend of mine was one on Saturday. The good news is that after waiting patiently, it will work.
As for purchasing the new iPhone, that has not gone well. Lines continue on Sunday to be a 4 hour wait in many locations. I have heard that there are problems with plan choice and credit checks, etc. Also 'famous' people have gotten to "cut" to near the head of the line.
Contrast this with the original iPhone rollout, a 2 hour wait the first day, no wait after that!
This product roll out has not gone well for Apple. They are not the first company to have these kinds of problems
Those upgrading the original phone should get a smoother interface, lots of little annoyances fixed, etc.