You need to look specifically for a camping first aid kit or put your own together. These aren’t the same as a regular first aid kit meant for urban living and scraped knees. Get a sturdy bag like a hard shell case or a backpack kept specifically for your first aid supplies.
The first thing you need to include is a first aid manual. Even if you know all about first aid, you might end up as the casualty and it will help someone else treat you.
You will need a range of bandages in different sizes, and absorbent gauze pads, also in a range of sizes: and burn dressings. You could include butterfly strips but you can also make these from medical tape, so include that. Ace bandages can be used for sprains and to fasten a splint or a sling if needed. Band aids are useful for small cuts or scrapes.
You will also need cotton swabs to clean any wounds and you may find it useful to pack a bottle of sterile water or antiseptic liquid or cream to prevent infection. Sterile water can also be used for irrigating the eye if anyone gets something in it. You will need scissors, tweezers for removing splinters and also a needle for those splinters that are too small for tweezers to get out.
Have a supply of gloves on hand to use while you’re performing treatment.
A flashlight would be useful, to see what you’re doing after dark. If you have children along with you, you might want a thermometer in case anyone runs a fever and appropriate medication to treat it. An ice pack would be useful to help with swelling injuries. You could also include antibiotic ointments and even allergic reaction tablets if you are going to an area distant from your usual home, since you will be in areas where you and your family may not have been exposed to the allergens. Pack the liquid kind if you have children who can’t swallow pills yet.
If you’ll be out in the woods you might encounter snakes. So take a snakebite kit. It’s a good idea to investigate the area you’ll be staying in to see what dangers you might encounter so that your kit can be built specifically for those issues, for instance, calamine lotion might be useful in case of light contact with poison ivy.
Make sure you have your health insurance details with you in case something happens that needs urgent medical attention.