i was going to post the same things that Cheshire Figment did - but he beat me to it while I was getting DD ready for bed.
The things I would add would be to check with your Pharmacist to see exactly what the storage requirements are as far as refrigeration. For some medications, it can be at a higher temperature for a short time, but should be stored in the refrigerator for longer times. The Pharmacist will be able to find that out for you.
Usually refrigerator temperature is about 35-45 degrees F. The Frio cold packs may work for short periods like between your room and the park, but may not get cold enough to store the medication all day. If that is the situation, I would use a case like that to transport the medication between the refrigerator in your room and First Aid in the parks. They will securely store the medication for you, as Cheshire Figment mentioned.
For timing, I would try to stay as close as possible to the times for doses aat home. We set a timer on our phone or iPod for DDs medication. Its very easy to lose track of time and miss doses of medication without using something like a timer to remind you.