There are a number of survival skills, knowledge of which would really help you in various SHTF situations. Not all of these involve firearms or other weapons at all but they are very worth learning, and a grasp of the basics is usually pretty easy. I have listed some below, I am sure you guys and gals can probably think of more of them:
First Aid - It is always nice to know how to stop arterial bleeding should you ever be so unlucky. Then again it maybe nice to know the difference between arterial bleeding and veinal bleeding. Of course, there are other more mundane wounds to which you may need to attend from small blisters, to sunburns, to bug bites - or even worse ones such as sucking chest wounds or a sharp stick in the eye. Take a first aid course - these are offered by groups such as the red cross. Any of these things could happen while on a wilderness hike, a family outing, or even at home.
CPR - Also offered by groups like the Red Cross, the pity is that you cannot perform it on yourself if needed.
The Heimlich Maneuver - This could be especially important is to know how to dislodge food that may be stuck in your own wind pipe should you be alone.
Basic Map Reading - Something I need to learn, although I can with a little difficulty read a topo to get me from A to B without too much up and down.
Compass Reading - Another thing in which I could use some lessons, although I have found my way with and without one many times before.
Distress Signaling - A good thing to know while on the hunting trip, skiing trip, or maybe even the family camp out when things go terribly wrong.
Swimming - what can I say about as basic a skill as this one.
Drown Proofing - it really could save your life someday if you enjoy water sports. You can learn this even if you cannot swim.
Foraging - I mean for edible vegetable matter such as plant leaves, roots, tubers, nuts, berries. (Forget mushrooms too chancy.) We all could probably use some more knowledge in this area.
Hunting - many people believe that having a gun and knowing how to use it is enough. You really should hone your hunting skills every chance. I live in an urban area, but sometimes try mine out on birds, or squirrels. Not as wary as those in the wild but not all that different especially since my dogs keep them pretty active in our small whenever they go out for a quick pee - see squirrel will chase. Even non-0hunters can ready themselves for a survival hunt by hunting with a camera and about a 200 mm lens. If you get fairly close shots, let's say of a deer, then you would be good to go with any rifle suitable for big game hunting with iron sights.
Then again there is the hunting of other critters that many would not want to consider eating. These would include: grubs, some insects, snakes, bird's eggs, crustaceans (such as crawdads), and the list goes on...
Fishing - None you don't just throw the hook into the water. Fishing is a skill that requires some knowledge to be more successful. Fishing can include use of hook and line, nets, traps and so forth. Yes you can even catch them by hand just like you have seen in the movies. I have done it, but it was not easy.
Trapping - In essence the same importance as the above two, maybe more important in the hunting arena, as it is a viable alternative or addition to any hunting methods.
Game & Fish Preparation - Ever ruin the meat that you were going to eat? I learned early on how to do that, so I also decided to learn right after that how to make sure to do it right. Ruining good venison is such a waste!
Cooking - Ever been on a hunting, fishing or camping trip and wish that Joe had come along because you remember that when Joe was there last year something was very different. Instead of eating spam right out of the can, Joe prepared dinner it so it tasted great, looked great and smelled great. Not bad to be familiar with the basics, it makes things a lot more pleasant when you can have an enjoyable hot meal so long as the situation allows.
Food Preservation - Know how to salt and sun dry fish for use later on. Food preservation could be handy if you come into a bonanza now and may face a shortage later.
Shelter Building - know how to build a lean-to that will not fall over in the first breeze that whispers Louise? Or do you know how to build a snow shelter that will help keep you warm? Get out camping with the kids and learn from them...
While I named a pretty good sized list of these skills, I am willing to bet you can think of more of them, I know I can.