I'm opening a business via the e-residency program with Estonia. I am a United States citizen and reside there, as well. Who should I file taxes with?
If the business is a limited company or something equivalent, and you are the one actually performing the work that the company sells, you'll most probably have to file taxes both in Estonia (because the business is registered there) and in the US (because you carry the work there). Where you pay taxes (and how much) is a lot more complex issue. There are exceptions to this, and lots of loopholes and tricks, but you really need to check with an accounting for relevant details.
You then need to file and pay personal income taxes (on dividends or wages paid by the company to you, or when you sell the company) in the US.
If you are operating as the equivalent of a sole trader or partnership, or some other forms of companies which are not subject to company tax, then the income is directly taxable in the US.
Some "light reading":
U.S. citizens, as a general rule, are subject to taxation on their worldwide income, but get a tax credit for taxes paid on account of that income to the country where the income is earned (subject to an exclusion from U.S. income taxation for certain wages and salaries earned abroad by natural persons). This can still sometimes be harsh because many countries raise proportionately more of their taxes from consumption based value added taxes, and proportionately less of their taxes from income based taxes on earning or profits, than the U.S.
The rules for determining when an entity is a U.S. person, subject to special tax rules because it is controlled by U.S. persons, or is a foreign person, are quite arcane. For example, consider the Internal Revenue Manual for Controlled Foreign Corporations which is just a small subset of these issues. https://www.irs.gov/irm/part4/irm_04-061-007.html