Howdy neighbor,
I'm a lifelong Raleigh resident and have traveled to Montana, Wyoming, and the Dakotas many times, including an out-and-back round trip over a 21 day period just last summer. Here are a few thoughts for your trip planning consideration:
Yes, 14 days is adequate, even for a solo driver.
Routing: From Raleigh, you may consider each of two routes from Raleigh to east of St Louis (Mt Vernon, IL). There are also two principal alternatives from near Omaha, NE to the Black Hills. In my opinion, the best route to Mt Vernon IL is I-40 to Nashville, TN, I-24 into IL, I-57 to Mt Vernon. The alternate is I-40/I-40 Business to US 52 in Winston-Salem, thence US 52/Future I-73/74 to I-77 outside of Mt Airy, I-77 to Charleston, WV, and I-64 all the way to Mt Vernon, IL. I favor the I-40 route since it's somewhat flatter and lacks the rough road and tolls along I-77 in WV. Nothing wrong with one route out and the other back. From Mt Vernon, I-64 to and all the way through St Louis to Wentzville, MO west of there, where I-64 ends at I-70. I-70 west to KC, beltway around the northeast side (I-435, I recall), and I-29 up to the southwest corner of Iowa, where the ideal alternates for the remainder diverge. Fastest would be staying on I-29 into SD, thence west on I-90 at Sioux Falls, SD, keeping I-90 all the way to the Black Hills. On the other leg, consider dropping down to the Nebraska Panhandle to Crawford and visiting Fort Robinson State Park. Fort Robinson is a well-restored US Army cavalry post. It features a large brick barracks structure built in the early 1900s and many single-family and duplex officers houses. Both the barracks and the houses are motel units now. From Ft Robinson, you may drop down to Alliance, NE to see Carhenge, a re-creation of Stonehenge only using old cars instead of stones. From Alliance, you may consider the "Sandhills Scenic Journey" along NE-2. NE-2 traverses the Nebraska Sandhills from NW to SE, passing through about 200 miles of unique 300-400' grass covered sand dunes and many small railroad towns. About midway is Mullen, NE, where you and the kids may stop for a half-day of "tanking", a float trip down the Middle Loup River in a 10' diameter galvanized steel livestock watering tank with seats installed inside. The Middle Loup is flatwater but moves fairly quickly and is a fun and relaxing way to see the Sandhills up close. At Grand Island, NE, get on I-80 to Lincoln, then connect to NE-2 again there for a shortcut avoiding Omaha to reach southwest Iowa @ I-29 once again.
Interesting sights along the way include the dams at Lake Barkley and Kentucky Lakes, near Paducah. A visit to the Gateway Arch in St Louis is always fun (but avoid the temptation to get on I-70 there, instead get back on I-64 to pass through a much less sketchy part of St Louis). There is a replica of the Lewis & Clark Keelboat at Lewis & Clark State Park near Onawa, IA, and it's only about 4 miles off of I-29. A large I-90 rest area near Chamberlin, SD overlooks the Missouri River (dammed up here) and has some good L&C history placards. At Wall, SD is Wall Drug, always worth a stop and a look around. Might as well spend a day on the loop drive through the Badlands NP, as well, as it's right there close to Wall, SD.
I'd start with the purchase of a Rand McNally Atlas, if you don't already have one. The above will make lots more sense with an atlas at your side.
Enjoy the planning and the trip!
Foy