Do I have to pay Federal income taxes from an inheritance (say $100,00)?

There is no federal income tax on the inheritances. Your state may have inheritance tax; check at state web site.
For more info read:
Your U.S. Tax Return: Tax on Inheritances

I am receiving $77,000 from my late Mom & Dads as the first installment of their estate. This just my share of the proceeds that came from the sale of condo, timeshares, and IRA accounts. We still have more to sell.

Do I understand that I do not have to report this as income to the IRS on next years filings?

I live in CA.

And, are my children, who will get $15,000 each, to do the same?

First, you shouldn't piggy-back a new question onto an old one, even though the topic may be similar.

Now, to answer your question - you do not report inheritances on your income tax. If the estate is large enough to trigger estate taxes being due (i.e. the estate value is greater than $2 million), the executor is responsible for any estate tax filings that may be due. Any estate taxes are paid out of the estate's assets, along with any other expenses the estate is responsible for, before the assets are distributed to the heirs. So by the time you receive your piece no further taxes are due. The one possible exception is that a few states still have an inheritance tax, which is paid by the heirs - but CA does not have one, so you are all set. Same applies to your children.