I'm still in the early stages but I have one piece of advice...

If your partner or someone else says "I'll do this, you go sit down/have a nap/relax" for goodness sake DO IT! DON'T try to be the hero supermum, you just carried a baby for 9 months, relax a bit! If bub is crying at night, and you've just fed him/her and your partner says "here, I'll try to settle him/her" for goodness sake LET THEM!
(because the helpfullness doesn't last forever )

I think maybe right now is a good time to talk about roles and expectations... I think you'll probably find you both have different expectations of each other, what you expect each other to do etc. Talking now would be better than snapping at each other for not doing this and that when you're both exhausted.

Thats all the advice I have, because I'm still living it myself.