First, they point out, the little guys are vulnerable and slippery--and I might add, based on my own child, may not love the water at first.
They need time to adapt--and for their skin to get used to air. Daily baths are not recommended for newborn babies. In between, you can gently wipe face and skin with a warm washcloth.
When you do dunk the whole baby--put it all the way in the water except for head and neck--this keeps them warm. Be sure water is below 120 degrees. We used to test with an elbow--why not?
Often babies will sort of go wide-eyed
Usually plain water is fine--if you must use a product, find one with neutral pH.
Diapers should not be left on more than four hours after being soiled. Clean the area with tap water. If you must use a wipe--find one without alcohol or lanolin. If a diaper rash develops--use Zinc ointment. This forms a barrier between the acidy stuff and the skin. If the rash worsens, ask the doctor.
I would add--always smile and talk to the baby while bathing. Make it fun and pleasant. For both of you.
PS You know all the stuff you get at a baby shower--one thing I really used was the bath towel with the hood in the corner waiting for a wet little head.