I am about to receive an inheritance (less than what is allowed before inheritance taxes apply). Do I still have to pay federal income tax on the money received? I live in Texas, where there is no state income tax.

No federal income are due on inheritances received.

The only taxes are estate (for federal) and inheritance (for the state) taxes, which should be determined and paid by the estate BEFORE you ever see the money.

Yes, any thing (money and property) you receive as gift or inheritance, you (the receiver) don't pay any federal tax.
Exception: If you inherit a traditional IRA, you are called a beneficiary. Beneficiaries of a traditional IRA must include in their gross income any taxable distributions they receive.

For any State tax liability check at your State's web site. For most of the States there is no tax.

MukatA's covered my backside once again: I forgot about the IRA exception.

Also, on the rare occasion where the executor screws up and fails to properly file an estate return when one is due, distributes the estate without paying the estate tax, then has the monumental bad taste of either dying or NOT having the assets to cover his malfeasance, then (and only then) will the IRS come after the heir(s) to pay the estate taxes that are due.

Finally, it is true that most states will not enforce an inheritance tax, because most states use the federal exemption guideline (which is about $2 MILLION dollars for 2008). However, that does NOT mean that the state do NOT have inheritance taxes. In fact, most states DO have an inheritance tax. It is rarely imposed because most estates are well below the $2M threshold. Further, if the estate pays an inheritance tax to the state, that tax serves as a credit against the federal estate tax, so the estate does NOT pay double taxes.

Is there a federal tax on a trust fund inheritance in Michigan

No INHERITANCE tax, but the estate tax MAY apply. It depends on HOW MUCH the trust is valued.