I am working in an old home and would like to replace a length of 3" Cast iron drain with ABS. What connector do I use to join the 3" Cast Iron to the new 3" ABS?

There is also a 3" copper line feeding into the original 3" Cast Iron at one end and I will need to join that end to the new 3" ABS as well. What joiner is recommended to Join the 3" copper to the 3" ABS?

Fernco shielded rubber coupling.
However I would use PVC not ABS.
Fernco 3 in. EPDM Rubber Shielded Coupling-P3000-33 - The Home Depot

Yep, agree on both counts with Harold

There is a properly sized coupling for just about every pipe transition, no matter what size it is or what material it is made of. You may not be able to purchase such coupling in big box stores, but if you go to suppliers that cater to professionals, they will have it in stock.

Milo