I have a full hospital, never mind first aid kit. Heck I even have a piggie medical table! That's what you get after two years of constant illnesses! rolleyes
For a basic kit, you'd want something to deal with physical injuries as well as illness. If you can have something simple to cover the most common illnesses you should build up a pretty rounded medical kit.
Physical Injury - Saline (or salt to make up salt water), wound powder, bandages, microporous tape.
Skin - Xeno, antifungal shampoo, Aloe Vera gel/ointment
Eyes - Optrex Infected Eyes or similar eye drops
Respiratory - Vicks/Olbas Oil
Digestive - Probiotic, Infacol/Gripe Water
Oral/dental - Daktarin Oral Gel, syringe-feed preparations (Critical Care/NutriPowder/Science Recovery) with mixing bowls, spoons and prepared syringes etc.
Pain relief - Calpol, Children's Nurofen (dose is 0.2ml twice a day)
Handy "tools" for the kit are syringes, a heatpad (e.g. Snugglesafe), towels, kitchen roll, latex/vinyl gloves. I also have a stethoscope, buccal pad separators and dental rasps.
Susie made a great point about keeping leftover Prescription Only Medicines. I have a bit of Baytril and a bottle of Metacam still. The only thing I worry about, particularly with medications mixed up by the vet, is how long it lasts. I go with a guideline of 6 months before I bin it.
Up to you if you worm your piggies or not, I do simply because of experiences with unexplained weight loss which were solved by worming. Hence I always have Ivomec and Panacur.
All of these would not be used to replace veterinary assistance, rather tide the pig over until you can get to a vet/seek their advice.