Bug Shield

Late to the party but I totally disagree that it is bad to shift both ends at the same time. In fact the rear, going to a bigger cog, tightens the chain. Shifting the front, which gives up chain length by going to a smaller ring, provides chain slack easier than the derailleur spring and will provide a better shift.

I almost never shift the front without shifting the back at the same time; On my bikes a chainring shift is about the same as two or three cog shifts, thus a shift on the front and rear gives a smaller gear ratio change and is typically what I want.

I do agree with the posters above who suggest that the derailleur/hanger was bent or the low limit screw was improperly adjusted and that spokes need to be replaced after being damaged by the chain.

PS. There are two ways to shift the front and back at the same time: Shifting for big changes (like the OP was doing) requires less chain from the derailleur spring. Partially canceling a ring change by also shifting a cog or two requires more derailleur spring but has the advantages of a smaller gear change and less soft pedaling than for two separate shifts.