I'm wanting to move my washing machine out of the basement to the next level upstairs. It's not far up and not far to the side of the original drain.my only concerns are can I use PVC and join it to the original galvanized drain? (Witch is vented straight up at the trap) and does it really matter how I run my new PVC drain? Through the floor and across the floor joists to the galvanized drain is how I planned? Should I use a rubber coupling? Any specs and suggestions would be greatly appreciated. Thanks!

ACROSS the joists or along a joist? Don't run drains through joists.
You can use PVC and rubber except under cast iron above, which is too heavy, and as long as you slope 1" in 4', you can put the washer further away from the stack.