He's starting to get pimples. We had thought that we only needed to bathe him once a week, though I can't remember where we read that.
Don't bathe small children too often, especially not daily! There's a risk they develop a skin condition like dermatitis.
In Austria, the recommendation from the midwives' association is to bathe infants weekly *' and newborns and infants ordinarily don't get so dirty that you need to use soaps (and lotions afterward) -- just let the baby's skin do what it's built to do.
*' And of course as needed! :-)
In reality, we bathed our son weekly while he was newborn, using only warm water, and that was fine. Now that he's 20 months old, we put him in the tub with intervals of roughly 3-7 days (and as needed when he's played in the sandbox). We let him play in the tub for a while, then use a soft cloth with a baby soap.
Can this be baby acne?
If it is, it has nothing to do with the number of times you bathe your child.
The rule of thumb torbengb mentions, to wash it once a week, is ok, altough I would bathe it more than that, but that's probably more because of social pressure and not for health reasons. We washed our baby regularly with a washcloth with lukewarm water and no soap. The Childcare organisation in Belgium does not warn against bathing with water per se, but it recommends to stay away from using too much skin products and soaps. It's better to use too little than too much.
We ramped it up. As an infant with sensitive skin, we bathed her about every 4-6 days or on an 'as needed' basis for those unusual incidents which require a little extra soap! We found that this was sufficient for general cleanliness. As she got older and more active we very slowly increased the frequency of baths. At 2 years of age now, we bathe her about every 2-3 days.
Note that when I say "bathe" I am talking about a light all-natural soap, a dot of all-natural shampoo and some frolicking around in the warm water - not scouring, harsh cloths or brushes or any of that nonsense.
"Cradle cap", if any, will fade over time on its own time - don't sweat it, and don't pick at it. Infant acne can be a perfectly normal reaction to the introduction of Mom's milk so don't sweat that either - this too shall pass. If it doesn't pass though, might be worth checking with a specialist.
Baby pimples actually have little to nothing to do with cleanliness. It could be baby acne. My baby had this for a few weeks, and we were told it was hormonal. The problem with bathing a newborn is that their skin is very, very fragile. This is even more true if you are dealing with a preemie. I would suggest sticking to once-per-week bathing for a while, and talking to the pediatrician if the acne gets particularly bad or you want to know more about when to start bathing more often.
Matthias got a bath once a week (barring diaper explosions) as a newborn with just plain water. "Baby Acne" could be his skin being a little oily, or it could be a result of the hormones from breastfeeding. As long as his skin doesn't feel oily or otherwise 'dirty', once a week for the first few months should be all that is needed.
My son is 3 now and the baths are coming daily, as he tends to bring the sandbox, garden, etc with him daily.