How about joining your local volunteer firefighters , or SAR and be authorized to operate on emergency frequencies for your area, as well as receive the training on radio operations used in your area as well.
I can and have contacted incoming emergency units as needed, but it is different when you are an emergency responder.
I think it has already been said, but I'll repeat it.
The cell phone call directly to 911 will be the best choice. If you do not have cell coverage, your best contact with an amateur radio is going to be to know your local repeater system. Let a base with a phone make the third party call to 911 for you. The repeater system will be likely to be monitored. Many are linked on the weekends to give a huge reception area.
Give the universal distress : "Mayday, Mayday, Mayday" . This will clear the frequency to your traffic and have all ears listening. Followed this with the nature of your emergency: "Traumatic injury , need medical help" Give a precise location. GPS Lon-Lat is great, but the county, nearest town , road names and nearest intersection will get ground units coming faster.
Contacting emergency units via radio should be a last resort. It will only serve to confuse in most cases. If you know the department protocols and frequencies that can help to limit the confusion.