It is strange, the bathroom fan is buzzing and humming when I first turn it on. After a few minutes, it starts up normally.
Why is it doing this?
Sticky old oil in the bearings. After buzzing (and probably turning very slowly) for a while, the motor heats up enough for the oil to liquify, and then it runs normally.
Try fresh oil if the bearings have a place to apply oil, or consider a new fan. This is a common problem with old fans.