Katiesmalls: ISO and JIS spindle tapers have been discussed a lot on this forum. Campy is ISO (or near-ISO) while Shimano is JIS. Square Taper Bottom Bracket Interchangeability

I would agree that the first step would be a good cleaning of both the cup threads and the BB threads. One useful trick to ensure you do not cross thread is to begin by turning the cup backwards. It should "drop" into the thread at some point, where you can then reverse to the appropriate tightening direction. If the BB threads are already cross-threaded, this might not help much. Close examination of the threads might show signs of cross threading?