and see that the higher income groups do, in fact, pay more in income taxes.
Except they don't. The problem here is that politicians are very sneaky, and it appears you've fallen for the trick: There is the Income Tax, and then there are taxes on income. They are not the same, but people (especially republican politicians) like to use them interchangeably. Most of the time, when a politician says "income tax", they are excluding the payroll tax, which is a big federal tax on all income... from wages, i.e. actually working.
To get an idea of the net effect of federal taxes, below are marginal tax rates for different types of income:
Income from wages: 25.6 to 43.6% (income tax plus payroll tax)
Interest: 10% to 35%
Short term capital gains: Same as interest.
Dividends and Long Term Capital Gains: 15%
Carried Interest: 15%
Tax-exempt municipal bonds: 0%
So while SOME of the "rich" (like doctors) pay a pretty high tax rate, most of the filthy rich ("investors") pay a very low 15% rate on their income. That's why Romney's rate is so low.
Again, many of the rich pay a much, much lower federal tax rate on their income than most of the middle class.