I think you need to check a map. The Shenandoah is in Virginia. I think that would be 'way too much driving for a week.
Where in Ohio are you coming from. You can be in the GSMNP from about anywhere in OH in a day. It is about 7 hours from cincy.
Asheville is not in the GSMNP--it is actually several hours from it.
Maybe really decide what you want to do.
How "senior" is your mother? Does she want to hike?
There is a lot to do. Hiking, wading in the streams, driving to overlooks, shopping in Gatlinburg, etc.
Asheville would also be a very nice stop, and taking the Blue Ridge Parkway north or south from there would take you to various places for some easy hikes and nature walks and waterfalls.
I think you may have the "shenandoah/skyline drive" mixed up with the Blue Ridge Parkway in NC.
I think you would have plenty to do and see and enjoy in NC.
Check back with some more info?
I assume you have some sort of a baby backpack for carrying the 2 year old?
When you ask "what gear should we bring", what is it you have in mind.
That time in September should be pretty good for reservations, and a pretty time of year to travel.

If you make Gatlinburg your first day hub, it's closer to Ohio than Ashville. From Columbus, Gatlinburg is under 7 hours. I would not break up a 7 hour drive, positioning you to take a hike or park activity your first day of arrival.

Gatlinburg is more touristy than Ashville, but within 30 minutes of several trails and amid Smoky Mountains best scenery.

Ashville could be your next hub. If you have not visited Biltmore House, it's a national treasure worth considering, but will take much of a day.

Another benefit to driving I-75 to Gatlinburg outbound is your return from Shenandoah will be WV Tpk and you will have different scenery on your return home.

Thank you Gretchen! Yes, I mixed up with the blue ridge parkway in NC, and you are right, Shenandoah is in VA.
I haven't done much research as you can tell...
I will be driving from Cleveland and stopped @ Cincinnati to visit a friend, then, we will continue to GSMNP.

My mother is 65 and she likes nature exploring too, waterfalls and moderate hiking will be great.
So if I skip the Shenandoah, and narrowed the trip to GSMNP and around area, am I looking at 3 stops then: GSMNP, Gatlinburg, and Ashville?

Will a week be able to cover these three areas? or should I cut down more?

Revised trip plan:
Day 1 - Travelling from Cincinnati to some place around GSMNP (Gatlinburg?)
Day 2 and 3 - Great Smoky
Day 4 and 5 - Ashville and Blue Ridge?
Day 6 - open
Day 7 - Back to Ohio

thank you stumpworks73, I like your 'hub' idea. do you have any recommendation on where to stay at these two hubs?

I like the northern part driving south from Front Royal VA.
Do you live in suburban Cincinnati?
I would head for Winchester VA the first day 7.5 to 8 hours.
Start early on day 2 to the Skyline Drive and cross over at Waynesboro onto the Blue Ridge Parkway. The Appalachian Trail crosses the SD and the BRP if you want to hike a short section.
Continue to get off the BRP at Roanoke.
Day 3 Roanoke to Knoxville TN. Day 4 Start early and see GSMNP.
Maybe walk another section of the AT. Do you have anything you really want to see in Asheville?
If you do stay in Asheviile, go back through The Smokies to see anything you missed and then head back from Knoxville to get home in Ohio via I-75. You could take a small detour in Kentucky to visit Mammoth Cave NP for a cool walk.

Mammoth Cave looks cool, thank you Tom!

If you're looking for hiking trails in the Smokies, here's a list of trails ranked by level of difficulty:
http://www.hikinginthesmokys.com/difficulty.htm

I'm sorry but Tom's itinerary makes zero sense. The first day is 500 miles of nothingness to get you the furthest point you can go.
You are a very easy 7 hour drive to SMNP from Cincy. There are number of threads with recs for where to stay. I suggest you look at the Park's site and decide what you want to do there. It is worth 2 days I'd say.
Then Ashville would be a nice stop--biltmore if you want. Otherwise it's a nice explore.
From there you could head north on the Blue Ridge Parkway. Take a look at their site for the various places to get off and "do" stuff. At some point going north, you can cross I77 and head north on that to go back to Cleveland without back tracking, and you will see some beautiful scenery in WVA.

Sorry I didn't catch the Cleveland start. If you have to see the friend in Cincinnati on the way, The GSMNP fist makes sense.
My route to Winchester would work if you could wait until the end of your trip to see the friend in Cincy.
In late May, we drove from Baltimore through Uniontown PA, Wheeling WV and up I-77 to Dover OH.
The next day we backtracked a bit and went to see a museum in Dayton and then went back through Columbus to get to Cleveland late in the day.
I think the Skyline Drive is nicer driven southbound from Front Royal IMHO. Knoxville TN is closer to the Smokies than Asheville is. I like the Blue Ridge Parkway north of Roanoke better than south of Roanoke (just personal preference).