As a lactation specialist, I work with a lot of new moms. I'd say, in your case, give your bf the first 6 weeks after the birth off, unless he's really eager and initiates contact. You could surely drop off a couple meals they can eat that night or freeze. New parents always appreciate a good homecooked meal. Check for allergies and preferences first (text yr bf to find out). Many breastfed babies are sensitive to cow's milk proteins their moms eat, so avoid dairy in case she's breastfeeding.
Personally, I wouldnt offer to do errands or babysitting until you know the mother better. I'd feel awkward if my husband's new gf offered to be in our lives in that way. When I'm newly post partum, I want old friends/trusted family in my life. Having her husband's new gf over might cause her some stress, and she doesn't need that with a newborn in the house.
Love withers under constraint; its very essence is liberty. It is compatible neither with envy, jealousy or fear. It is there most pure, perfect and unlimited when its votaries live in confidence, equality and unreserve. -- Shelley
me: Mags, 59, living with:
miss pixi, 37