chicken our NCT teacher was v encouraging for dads to do skin to skin after the birth. After DD was born, I held her and fed her, then he put a nappy on her, took off his shirt and cuddled her for ages. I was pretty tired so glad to hand over for a bit.
Absolutely agree re soothing of unsettled baby. if I'd just fed DD and she was still crying and unhappy and we'd done the usual winding and nappy change then he had more chance of soothing her as he didn't smell of milk, which was confusing to her sometimes I think. As in she then thought she wanted milk, so I offered it, and then she drank for two mins and then howled some more.
I expressed and DH gave her a bottle at 10pm (from about 3 months in) and then I could go to bed at 830 and sleep through til she woke at 2 or 3. it made a huge difference.
I have a friend whose husband used to do the night shifts on weekends. So when the baby woke, he'd bring it to her, she'd feed and go back to sleep, whilst he'd do the changing of nappy, settling etc (depending on the baby this can be an extra 30-90 mins) whilst she slept. He then brought the baby in to her at 6 or 7 and had a long lie to recover.
I bought DH the Fatherhood book by Marcus someone and it was great for preparing him a bit.