Now that I'm in the clinical setting, I will say that getting experience in treating adult patients with common conditions (e.g. diabetes, heart disease, COPD, etc.) in the hospital setting was very beneficial before starting my NP program. By doing so, you become familiar with the medications prescribed, common lab values, and more importantly, the physical signs/symptoms you see with these patients. As an NP student, I often have to do assessments on new patients to the practice. Knowing what to look for in a physical assessment and having my experience at the bedside has helped me countless times in recognizing the need for further diagnostics, such as EKG or PFTs.
If you are going to specialize in Peds, you might want to consider being a PNP, instead of FNP, if you don't want to work with adults. The biggest drawback for me in clinicals is that my only experience with Peds & OB/Gyn was during nursing school. Never had to deal with it once I started working in the hospital. So, if you want to have a well-rounded experience before starting your NP program, it would definitely be beneficial to gain some experience with adult patients. You don't necessarily have to work in a hospital setting, if that seems daunting to you. Perhaps a sub-acute rehab or LTC facility? Some adult specialty practices hire RNs (e.g. cardiology, pulmonology, etc.) so you might be able to find an outpatient practice. Look at what's available in your area.
Good luck with your decision