If my son was bitten by a neighbor’s dog in the neighbor’s yard, how should I go about getting medical expenses paid for?

When your son is released by the doctor, obtain your son's medical bills and medical reports (I assume that your son is a minor and there isn't any wage loss claim).  Your son's personal injury claim should include the medical bills and medical reports and should be filed with the neighbor's homeowner's insurance carrier.  Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of your son's injury and will be used to determine compensation for pain and suffering, which is an amount in addition to the medical bills.  If the case is settled with the neighbor's homeowner's insurance carrier, NO lawsuit is filed.  If you are dissatisfied with settlement offers from the insurance carrier, reject the settlement offers and file a lawsuit for negligence against the neighbor.  Negligence is the failure to exercise due care (that degree of care that a reasonable dog owner would have exercised under the same or similar circumstances to prevent foreseeable harm).  If the dog has bitten others in the past, then your lawsuit against the neighbor would be for strict liability instead of negligence.  Strict liability imposes liability whether or not due care was exercised.  If your son is a minor, you will need to be appointed guardian ad litem to file a lawsuit on his behalf because a minor cannot file a lawsuit himself.  If the case is NOT settled with the neighbor's homeowner's insurance carrier, your son's lawsuit must be filed prior to the expiration of the applicable statute of limitations or your son will lose his rights forever in the matter.