Yellowstone, Mount Rushmore, Grand Teton, Custer State Park.
 

Planning at trip the middle of October to these places. We will be flying from Texas, renting a car to drive around once there.
Any suggestions what city to fly into and what city to leave to fly back to Texas
We are planning about 10 days but can be shorter or a little longer
Also suggestions for lodging and any other helpful suggestions.

Thank you

You might want to review the link below regarding Yellowstone road closures. Your only safe bet is to enter/exit Yellowstone the only road that is maintained in Winter.

http://www.nps.gov/yell/planyourvisit/hours.htm

A lot of lodging options will be closed at that period in Yellowstone/Grand Teton, but less circulation.
Jackson Hole is a good basis for Grand Teton(out of park but not so far).
For Yellowstone, Gardiner just north of park and West Yellowstone are 2 possibilities, but you will have to drive some miles to be in park and the to the "attractions" and effectively some roads are closed.
Erik

I suggest that you not miss Badlands National Park. Fly into Rapid City in the early morning and drive to BNP.