On the whole I am not a fan of supplements, you can get all that you need from a well balanced diet, however there are a few exceptions.
This is the main supplement that you should add to your diet, because of changes in climate I don't believe it is possible to get the amount of vitamin D we need anymore. I tell my kids it's the sunshine vitamin, because that is where we get most of it from, it is found in small quantities in some foods like fish and egg yolks, but you can't get what you need from diet alone. It is also produced by the body in response to the skin being exposed to sunlight. The problem is that we don't spend as much time outdoors as we should and we are a lot more clothed than nature ever intended, which reduces the amount of sunlight we get.
Vitamin D deficiency has quite sever health implications. Calcium requires vitamin D to be used effectively, so an insufficient amount can lead to weak bones and osteoporosis and it is also associated with a higher risk of cardiovascular disease, asthma and cancer.
Thankfully there is a vitamin D supplement you can take and its fairly inexpensive, just make sure that it is vitamin D3 not D2, as D3 is the type that is produced by the body in response to sun.
I take 4000iU and give the kids half of of that.
Magnesium is probably the second (after vit D) vitamin/mineral we are most deficient in and is important for lots of things such as, nerve and muscle function, immune system, strong bones and also helps regulate your blood sugar. Again it is something that we don't get enough of because of modern lifestyle. We used to get enough from the food we ate, but because of mass farming the quality of the soil fruit and veg is grown in has decreased and it no longer contains the high level of vitamins and minerals it used to.
Deficiency can lead to a whole host of health problems such as nerve problems, osteoporosis, depressions, blood clots, diabetes, heart/kidney/liver disease and panic attacks.
You can supplement it in two ways, as a tablet or as liquid or flakes that you can add to baths or yourself after a shower. Personally I use the liquid form two to three times a week. It feels a little tingly at first due to the concentrated dose, but you'll find this goes away quickly.
Omega 3 is an essentially fat (meaning it has to come from our diet) that we get from food such as beef, fish, some nuts and seeds. Deficiency is strongly linked to heart disease, high cholesterol, high blood pressure, diabetes, osteoporosis and depression.
Due to quality of our food, most people do not get as much as they need. You probably know that oily fish such as salmon, tuna and mackerel is high in Omega 3, but you may not know that this is only true if the fish is wild caught. This is because of the fishes diet. In the wild, fish get there omega 3 from the food chain. Algae, high in omega 3 are eaten by krill, krill are eaten by small fish and small fish are eaten by larger fish, so the omega 3 is passed on. This doesn't happen in fish farms because of the processed food they eat, so they have almost no useable omega 3. The same is true for beef. Beef should be quite high in Omega 3, but because most of your meat found in supermarkets and butchers has been fed on an unnatural diet of grains, it has changed the fats contained within the meat. Grass fed beef contains twice the amount of omega 3 than your standard grain fed cow.
It isn't necessary to supplement omega 3 if all your beef is grass fed and all your fish is wild caught, but the cost of eating like this will be a big barrier for people, because it is not cheap. It's also not going to be possible if you are vegetarian, so I would recommend supplementing with 4000mg a day from either fish oil/capsules or flax oil.
Vegetarians are also going to struggle to get enough iron and vitamin B from their diet, which us herbivores get from our meat, so they would be well advised to use a supplement like Flovital. 20ml for adults and 10ml for kids will provide what you are missing.
Apart from these three (four if your a vege) things you should be able to get everything else from a well balanced diet, high in quality vegetables, meat, eggs, nuts and seeds. Just remember that quality is the key word here. Eat as much organic veg, grass fed beef and wild caught fish as you can afford, you will be healthier for it.