I'm traveling and I don't have access to my manual, so I'm sorry if it's in there. I flip between modes a lot and tend to use full auto (when family doesn't have patience for me to set something up). I've noticed, however, that the camera reverts back to single shot from continuous. Is there a way to force it to always stay on continuous mode? All google searches just had how I've been setting it through the shooting menu.
As you seem to already have figured out in the comments, Full auto forces a number of settings on you. Many cameras allow you to create your own programs, so if the D5100 has that feature, you could create a Full auto program but with continous shooting.
If you only switch between P, A, S and M, the drive mode should stay unchanged. Practically everything can be achieved in these modes (hence you may not need the auto modes), it just may need a bit more time for setting up for some scenes.
I don't use Nikon, I have Canon, but this tip may help. On Canon, the camera switches to Continuous shooting automatically when you use Scene/Portrait mode.
Usually, when you change that in the "<-i->" menu, it stays in that mode... Maybe the auto program is changing the shooting mode? That would be the only thing that explains that for me.