oblige (one) by (doing something)
To make one indebted by doing something for them or on their behalf; to do something that accommodates one. You're under no circumstances required to oblige your employers by giving up your legal rights in this case. I was wondering if you would oblige me by picking up a parcel for me from the post office.
oblige (one) to (do something)
To compel, bind, or otherwise force one to do something out of a moral or legal obligation. You cannot oblige your employees to put themselves in harm's way. It's simply not the case that pregnancy obliges a couple to get married any longer.
oblige (one) with (something)
To grace, accommodate, or provide one with some service or favor. The famous movie star was always happy to oblige his fans with autographs and pictures. Go on, Auntie May, oblige us with a story!