Teach them that taking responsibility means doing what needs to be done, not just what is fun to do or what you want to do. Taking responsibility means not waiting around for others to do it for you. Taking responsibility means not assuming if you don’t do it then someone else will. Take the initiative and do it yourself.
2) Strong Work Ethic -- Having a strong work ethic is a major key to success in life in any situation, not just in survival situations. Teach your children the value of hard work. Hard work is good, healthy, and effective. Its importance is revealed throughout God's Word. The idea of "an honest day's work for an honest day's pay" is part of the traditional American value system.
Children, even young children, should have chores that they are expected to do. And make them actually do their chores. The chores should be age-appropriate, but their responsibilities should increase as they grow older and gain maturity and skills. Older children can work part-time jobs outside the home. You are not helping your kids by shielding them from hard work.
3) Biblical and Traditional Values -- Teach your children to have a relationship with God. Make prayer, scripture reading, and worship a regular part of their lives. This means it will have to be a part of your life, too. Teach them the Commandments and the teachings of Jesus. Teach them to look to the perfection of God's Word as the ultimate authority for what is right and wrong, instead of the whim of worldly opinion. Don't just teach them what you believe, but why you believe it. Equip them for a world that is telling them that Biblical and traditional values are wrong.
Also, teach them traditional American values - including the importance of individual freedom and self-reliance, the importance of our Constitution and the Bill of Rights, the importance of private property rights, the moral superiority of real capitalism, the value of hard work, the value of human life, common decency towards others, and the traditional monogamous family unit as the basic building block of society. Be sure to teach them that our rights come from God, not from government. (If you have any trouble explaining these concepts to your kids, it is likely because you were never taught them. This intentional dumbing down of America has been going on for decades.)
4) First Aid (including knowledge of wild medicinals) -- If your children are old enough to be in school, they are old enough to learn basic first aid. They can advance their skills as they grow older.
I would also start teaching them early on how to identify wild edibles and medicinals, adding to their skill sets collecting, preserving and using them as they mature. Consider having them start and tend a medicinal herb garden. Again, start with the basics, and help them grow their skills as they get older.
5) Self-Defense Skills -- Teach your children situational awareness, which is more than just paying attention to what is going on around you, though that is an important start. It means both knowing what to look for, and how to assess (make decisions about) your surroundings. Check out my article on Situational Awareness and start teaching your children these skills.
I also urge parents to enroll your children in a good martial arts class. Learning a martial art such as karate or judo can be a fun hobby, provide considerable health and fitness benefits, improve self-confidence, and is a life-long self-defense skill set.
If your family or group has guns, then your children need to learn gun safety at a very young age. Your local gun store may be able to point you towards age-appropriate gun safety courses. Teach your children to respect guns, not to fear them.
It is up to you to judge the maturity level of your children and decide when they should start firearms training. I grew up around guns and was shooting and hunting with my grandfathers by the time I was 8 or 9. They were with me to provide guidance and supervision, but I was carrying and shooting my own gun (a .410 shotgun). I also had to help clean anything I shot, even at that early age.