We are doing this already.
My sister is a med/peds doc so once she's done with residency (one more year, yay!!), she'll work as a general practitioner in a family medical clinic. She basically specialized both in internal medicine (grown ups) and pediatrics (newborns through teens). However, many places don't have people that do both so it pays to check.
The nice thing about being seen in a clinic like that is that if your entire family is down with the same bug, you can all be seen by the same physician rather than going to separate clinics for the same problem. AND - if you find one you like, your LO can be seen by the same doctor the entire way through their life from baby to adult which is really helpful if they develop an allergy or some sort of complicated medical issue. My sister said that one of her clinic patients was so grateful for that continuity of care for her son who has autism because once he got comfortable with his doctor, they didn't have to keep changing it up as he got older.
As for questions, I think that depends on your beliefs regarding vaccination schedules and such. If you want to pursue some sort of alternate schedule or treatment plan, some docs won't keep you at the clinic due to liability concerns. If you plan to go with the prescribed regimen, then it really is a matter of personality I think.
We are going to use the same doc my sister uses for my nephew (it is complicated for docs to see family members as patients). She is fairly young so would be up on current issues, and is HIGHLY regarded by the resident doctor staff which is where my sister knows of her. Bonus for us, she's in a clinic attached to a hospital so if there is ever anything seriously wrong with LO and they need to be hospitalized, LO can be seen in the hospital by his regular doctor.