Hmm... I'm not looking forward to interviewing pediatricians. I don't even know how to find a doctor for me!
The things I'd ask about are vaccines (I feel very strongly about vaccinating, and I don't want a pediatrician who doesn't agree - I especially don't want to bring my too young to be vaccinated baby into a waiting room full of un-vaccinated kids), and antibiotics. My mom was very firm that we only got antibiotics when we actually needed them. None of this placebo, please do something for my baby stuff. I think she has the right idea, there are all sorts of downsides to overuse and unnecessary use of antibiotics, and I don't want a doctor who hands them out every time baby has a sniffle.
Also, general bedside manner - does he seem like he'll be good with kids? And is he understanding about stupid questions from first time moms? Do you have a similar philosophy about when to bring baby in? Is there a nurseline or something you can call if you have concerns? Are the office hours good? Would you be able to get a last minute appointment if the baby is sick? Also, as your kid gets older, gender could be an issue - She may not want a male doctor examining her.
As for daycare, we're not doing it, so I only know what other friends have discussed. Some things I've heard about are baby to staff ratio, extras like foreign language lessons (obviously with older kids), their sign in/sign out policies (and do they really follow them?), and what would make them kick a kid out? What's the policy if your kid is a biter? Or someone else's kid is biting your kid? Potty training? Is there an age cutoff where you'll get booted if the kid isn't out of diapers? Late pickup charges? Sick kid policies? And also, do you like the staff? Do they seem nice? Like they actually like babies?