Do the same pilots fly connecting flights?
It depends on what you mean by connecting. I don’t think you’re using that term the way you think you are. So I’ll just say that typically two pilots are kept together during their trip unless one or both of the pilots have managed to drop or trade portions of their trip with someone else. There are other scenarios that could lead to a crew change but typically pilots are paired together and stay together.
What do you mean by “connecting flights”?
If you mean your flight goes from say EWR through BNA and continues to IAH, rather than direct EWR-IAH then yes.
But not always. Flights from SEA to OGG connect in PDX, ANC, LAX, PDX and SMF (maybe others). When I travel home to SMF the plane continues to OGG with a new crew.