Pretty much every source I've read recommends keeping two to three weeks worth of prescription medication on hand for emergencies. I have to tell you that I find this to be pretty ironic. In this day and age of exaggerated concern over prescription medication abuse it is almost impossible to get a doctor to write a script for extra medication. Or to get a pharmacy to fill it, and last but not least, to get your insurance company to pay for it. I have managed to build up a small extra supply by refilling my medication the three days before it runs out that my pharmacy and insurance company allows. This means three extra pills a month. The medicine supply does build up slowly this way.
Next I would recommend a comprehensive first aid kit. Not one that is just for cuts and scrapes, but a kit that has splints, suture materials, blood pressure cuff, etc. There are many good prepacked kits available online, but I would encourage you to personalize it to meet your family's needs. Consider throwing in some super glue (replaces sutures), anti diarrheal medicine and a ton of sunscreen. Research what is recommended for your geographical area.
First aid and First Responder training is available through the Red Cross and some schools. I also received my certification in Advanced Survival Swimming because I live on the Florida Gulf Coast. Think about the type of medical training that would apply to your family and area.
Oh, and don't forget your pets. Ask your veterinarian what they recommend you keep on hand for emergencies.