I'm a U.S. citizen who lives in the U.S., but I'm going to be working in Mexico for a Mexican company. Do I still have to pay taxes in the U.S.?
Since you are a US citizen, you have to pay US income tax on your world-wide income, but you do get to deduct (or take a tax credit for) taxes paid to other countries from the taxes you owe to the US. Tax treaties also affect such matters. Also watch out for reporting requirements for foreign accounts that you have, or will likely have once you start working in Mexico.
In addition to the Foreign Tax Credit, which @DilipSarwate mentioned, if you stay out of the U.S. for more than 330 days of the year, you can also use the Foreign Earned Income Exclusion, with which you can exclude your first ~$95K of income from U.S. taxes.
In any case, you will still have to file U.S. tax returns, regardless of whether you actually have to pay any money.