To be a bit more specific, the taxes you pay on reinvested dividends or capital gains are NOT added to your cost basis when you sell - just the reinvested dividends or capital gains.
My question really was will I somehow be taxed on my cap gains twice. I spoke to another expert who said that my cap gains will up adjust my basis So that I am not taxed on it twice. Does that make sense?
You will not be taxed on your capital gains twice. You will be taxed in the year that the capital gains are received. That amount of the capital gains is added to the basis of your investment and when that is sold you will be taxed only on the gain, not the basis.