I've heard it said that as long as you eat a balanced diet, you don't need extra supplementation. In a Utopian society, this might be true. And outside the discussion of what a proper diet is, isn't truly feasible.
The first problem is with the quality of the food that we take in. Most mass produced foods have been selectively bred to maximize production at the expense of nutrition. On top of that, our food is typically shipped from other states and other countries. And aside from the environmental impact, to ensure that the produce is ripe when it reaches the stores, it is often picked while it is still green. Not allowing fruit and vegetables to fully ripen while in the soil means, that it isn't going to produce all the beneficial chemicals or absorb all the beneficial minerals that it would otherwise.
The same can be said of our animal products. Animals are bred and fed to grow large fast, and are often pumped with chemicals to either grow faster of protect them from the living conditions that allow more animals per square mile to be raised.
You can eat organic and local, and you can eat pastured and grass feed animals. That helps A LOT towards getting all the nutrients in your diet that you need. It sill can't account totally for the state of the soil and the breeds of plants and animals that are around these days, but it does go a long way.
The other consideration you need to take into account is your activity level. The nationally set Recommended Daily Amounts (RDA) are for someone with a relatively sedentary life. If you are reading this blog, more than likely you are more active than your average person. In that case, you're going to need more than the RDA to attain optimum health.
So yes, you are going to need to supplement your diet. The first place to start is a simple daily multi-vitamin. They usually have 50-100% of the RDA for many things, this will fill in many gaps and your body will pretty much flush out what it doesn't need. These are cheap, and the drug store brands usually suffice.
Another necessity is an antioxidant. This is a huge area, with a lot of science behind it that will eventually be its own post. For now, a few that I recommend are Vitamin C, N-Acetyl Cystine (NAC), and a vitamin B complex (which has the added bonus of extra energy).
If you want to go beyond the basics, I recommend the book "Sports Nutrition Guide" by Michael Colgan. In this book, he details all the vitamins, minerals and antioxidants that athletes of various sorts need, how they help with metabolism and strength, and in what quantities one should have them (and if one can possibly have too much).