What would the benefits be if I were to pursue a career in the Air Force as a pediatrician?

What is your definition of Benefits? When speaking in military terms, the word Benefit could have ambiguity. If you are asking what are the military benefits of joining with a degree in Pediatrics there are a couple of routes. You could join as an enlisted member for a minimum obligation of either 4 or 6 years( this may have changed as I have also heard of 3 year commitments in the past). Here are the pros and cons:
Pros:
-Shortened term of obligation (in case you don't like it)
-Experience
-Possible faster promotion due to your degree

Cons:
-Starting at the bottom
-less pay
-Extra training ( that you may have already received)
-Staying on base in dorms/barracks (if you are not married)(this may be a pro though: saving money)

Benefits:
-Military pay to include: Housing allowance, Sustenance allowance, clothing allowance
-Free Medical
-Free housing
-Travel (Space Available allows a very discounted airline ticket with military aircraft)
-Tax free shopping at the exchange
-Commissary access (tax free but charges a surcharge - which is like taxes)
-Tuition Assistance (if pursuing graduates)
-Free training
-Ability to apply for role changes
-Multiple Military clauses (leasing a car, buying a car, rent, Cell phone/Cable contracts, etc.)
-Tax free retirement(thrift savinigs plan) *I have heard this might change to a 401K type of plan
-Early retirement age (only obligated 20 years to start receiving retirement pay) the more years you do, the more percentage you get in retirement pay

If you were to join as an officer here are the pros and cons:

Pros:
-Better pay
-Better positions to lead/manage

Cons:
-Working on commission (length of term varies, some are as long as 10 years)
-More responsibility
-More training (Officer Training School)
-Deploy more/Expectation to deploy is higher
-Easier to voluntarily separate (this may be a good thing though)
-Expectations are higher - punishments are harder

Benefits:
-Same as Enlisted as far as pay, medical, etc
-Better housing accommodations (not required to live on base in dorms/barracks)
-Higher retirement pay

IMO, joining the military would be a great experience - whether your time was bad or good, you have something not many people can say they have experienced. It is a culture that not everyone can get on board with, but it does define things in your life - helps you realize what you really value in life and how you work your life around those values, how you seek those values in others. What I tell other about the Air Force is that you may not be challenged physically, but it will definitely challenge you mentally. You will be asked questions that there are no answers to and expected to find those answers. You will be expected to think for yourself and some decisions you make will hurt and affect others lives, and you have to find a way to morally come to terms that your decision is/was correct and if it isnt there is no other way to get passed it other than to accept it.

You are quite a distance from becoming a Pediatrician in the Air Force...at least 8-10 years. Probably the biggest benefit of being a physician in the Air Force is the ability to provide care for your patients without having to worry about if they have insurance, if they can pay for care, of if you can get critical care "approved". Look at the avenues available to become a Pediatrician (4-yrs Pre-Med, 4-yrs Medical School, 4-yrs Pediatric Residency), the Air Force offers many opportunities for enlisted members to pursue their goals. Take a look at the following two links...one connected to the Air Force Medical Service and one to the Recruiting site.

http://www.airforcemedicine.af.mil/About-Us/

www.airforce.com