Recently both my parents passed away within two days of one another. Each had their own retirement plans, life insurance policies, and my father had a substantial amount in annuities which he received tax free. The administration is set up to where my brother and I (heirs of both my parents) are to each receive 50% of my father's estate, annuities, and insurance policies. Also, my sister (heir of just my mother) is accepting dispersements from both my bother and me (we wanted to include in her but the laws in Texas will not allow her to be a beneficiary/heir of estate or other policies since she was not a true heir of my father). What I need to know is what is considered taxable income on my part and what is not. I was told that a general rule of thumb is, whatever my parents would have considered taxable income so should I when I inherit it. Otherwise I should not report it as income on my tax returns. Is this correct?
Also, will my half sister be taxed on anything my brother and I give her as her "share"?
I appreciate any helpful responses.

First, sory for your loss. Losing both parents so quickly must be a real shock.

Please verify that your parents died without having wills, and that's why the state of Texas laws are affecting your half-sister.

Your inheritance is not at all taxable to you. Neither TX nor the federal government have an inheritance tax. If the estate owes any estate or gift taxes the executor would pay that out of the estate assets before distribuing the remainder of the estate assets to the heirs. So no - you do not need to worry about reporting any income on your taxes due to your inheritance. However, once these assets are yours and start generating income for you that income is likely to be largely taxable (it depends on the specific investments).

As for giving some of your inheritance to your sister - because your inheritance belongs to you, if you transfer some to her it is considered to be a gift from you to her. Hence it is not taxable to your sister, but if it is in excess of $13K you will have to file a gift tax form with your federal income tax to document the gift. No actual tax is due unless the total of all gifts given over the years that exceed the annual threshold adds up to more than $1 million.

One last point - be sure to ask the executor for records on the cost basis of the assets in the estate, because you will need this information if/when you sell any of the assets, or gift them to your sister. If your parents died in 2009 or earlier the tax cost basis is the fair market value as of the date of death (or optionally 6 months later - it's up to the executor to determine which). If they died in 2010 the cost basis is equal to your parents' original cost basis. Obviously this may be difficult to determine, but hopefully your parents left very detailed financial records going back years. Again, ask the executor about this.

The will issue is most important, because if the will of one parent made the other parent sole heir, that transfer may cause you and your brother OR your half-sister to lose their inheritance, because the first-dying parent's estate effectively became part of the second-dying parent's estate unless there was a provision in the will that requires the second parent live long enough for the will to be properly probated.

BOTTOM LINE: If there are wills, RUN, do not walk, to a competent estate attorney to get these issues settled.