The type of doctor you see is entirely up to you. Different people want different things from their physicians. I've had luck by asking people I really trust for physician recommendations. As another poster noted, many practices will allow you to tour the facility and meet some of the staff before becoming a patient there. That can help you get a feel for the place and the staff - you can get a sense of whether you're comfortable there.
You might want to inquire about policies or staffing matters that concern you. For instance - does the facility have same-day appointments for illness? Does it have any evening or early morning hours? Will you be seeing a physician each time or will you sometimes be assigned to see a nurse practitioner or physician assistant? (There are good and bad aspects to both possibilities: being seen by NPs and PAs usually means you can get in for an appointment in a timelier fashion, but you might have a harder time developing a relationship with a specific provider.) Will you have any online access to your medical records with the practice? Can you contact a physician or NP by email with questions or concerns? Is the practice associated with a larger hospital network? Is the practice considered in-network by your insurance company? Also, it never hurts to check on your prospective physician's background - you can look up details like date of graduation, medical school, and any formal actions taken against the physician through your state's licensing board website. You can search for and verify your prospective physician's board certifications (for free, though you'll have to register) through the website of the American Board of Medical Specialties: https://www.abms.org/
As for your records, if you're going to see a new physician because of a specific health problem (as opposed to something like an annual checkup), it would probably be a good idea to have your records sent ahead once you choose the practice you want to use. Contact your pediatrician's office and ask for whoever handles their medical records requests. They'll have you sign a release form before they send the records anywhere. You should be able to fax it to them if you live outside the area, and some places will allow you to scan in the signed form and email it to them. If you have had any major surgeries or were hospitalized for any major illnesses, you might want to contact the hospital(s) where you received care to have those records sent on to the new physician as well. Ask for the hospital's health information management or medical records department. (In theory, most of your records from any hospitalizations should be in your chart from your old pediatrician's office, but theory and reality often differ quite a bit.)
For the future, it might help to look into one of the personal health record (PHR) options that are available now. Google Health and Microsoft Health Vault are two good and free options that allow you to keep and organize your own health records; however, there are many different PHR options out there. Some PHRs will also automatically add in new data from sources like certain pharmacies, hospitals, and lab test companies, which is very useful. You can also do things like track weight loss or metrics from various health conditions (like blood sugar readings for diabetics), and measure your progress towards various goals you set. Some PHRs allow you to set them up so that your physician can access your data (obviously, this is something you would have to specifically set your PHR to allow). They can be very helpful in a lot of different ways. So that's just an idea for you to consider and research if you choose.
I apologize for the length of my post. I hope at least some of it helps. Good luck to you!