I'll try to help!
Private vs. public: In a public (or county hospital) type setting, you can expect more a more indigent population, perhaps kids who haven't had excellent primary care as of yet, immigrant kids, etc. In the private world, more pts will have insurance, may be more likely to have had good primary care from the beginning, etc. As far as how busy each is, I think either one could be just as busy...it mostly depends on the individual clinic. In general, I would guess that public settings may have somewhat more pt volume, but that's not necessarily true in all cases.
Solo vs. small vs. large group: These practices will basically differ from one another in how things are done day-to-day, especially with respect to who is responsible for pts who get admitted, who's on call, etc. Larger groups, since they have more people, might share call equally, so that within a 5 person group, each person is on call every 5th weekend (or something like that...just a general example). Also, in a group practice, someone different might see "sick visits" on different days, or might be in clinic for sick visits one Saturday every few weeks. On the other hand, in a solo practice, that physician is responsible for all of those things.
Urban vs. Suburban: If you're in a general pediatrics practice, this won't make a whole lot of difference. I think it more depends on what other children's services you have in the city. For example, where I am we have a large children's hospital, where a lot of kids are seen...and there are also a lot of general pediatrics practices in the suburbs, and also some in the city...
In choosing where I wanted to have my outpatient residency experience, I ranked private practice very highly, because that means I'm more likely to be assigned to a true general pediatrics practice, which is what I ultimately want to do. I did not rank the outpatient clinics in my Children's Hospital as highly because I have heard that the patients seen there tend to be the complicated, chronically ill kiddos who also have a few specialists they see on a regular basis. As a general pediatrician, sure, I'll have some complicated patients, but I want to make sure that I learn how to deal with well children and their runny noses, belly aches, etc.
Hope that helped a little bit! You will catch on to "the system" as you begin your training here in the US...it can be different from the way things are done in some other countries, but not too tough to learn, and I bet you'll catch on quick.