If it is Hartford CT that you are starting from, the place you should stop is either Winchester or Front Royal Virginia.
Depending on the month of your travel, you could take a drive on the Skyline Drive getting back to I-81 via Waynesboro and Staunton.

We are assuming here that you are going to take I-81. Is this true? There are other ways to get there from Hartford, obviously, and some of those other ways have some interesting possible stopovers.

Yes, we'll be taking 81. What is in Winchester and Front Royal? Thanks

Front Royal is the north end of the Skyline Drive.
Cut back over to I-81 at Staunton.

Winchester is a largish town with an ample selection of the usual motels and hotels. Lacking in charm but convenient.

Front Royal/Skyline Drive is pretty and fine in summer, but it is off your route and slow (35 mph speed limit in many (most?) places, a problem if you are trying to get to Asheville rather than see the country. If you are a Civil War buff, US 11 takes you through many sites on a route parallel to 81.

Staunton and Lexington are both worth stops.

The whole trip takes 12- 13 hours, so you could do it in one very long drive. I would want to stop for the night along the way though. Since the first part of the trip is the most stressfull to me, I would want start very early from Hartford and make a break before half way. Google maps takes you through NY city then down to pick up 80 west to 81 south, but we usually take 84 west to 81, or sometimes 84 to a drive south on 209 through the Delaware Water Gap to 80 west to 81 south. If you do the NY city thing, leaving Hartford by 4AM puts you through the city by rush hour.

I would make a stop at Gettysburgh or Harper's Ferry for a couple of hours and then stopping for the night in that skinny part of West Virginia or very northern Virginia. I would spend a little time on Skyline Drive the next day and then switch over to 81, although if it is winter Skyline Drive may not be an option.

I think I missed the part about "cool". I think I missed it because there isn't much along this route that could be called "cool" at least in the hipster/Brooklyn sense. People may look at you funny if you dress in black and wear Daniel Liebeskind glasses.

But Harrisonburg is not far beyond Winchester and at least has a university (Jame Madison) and various student hangouts and music venues. It should not be a problem during the college year to find blues, bluegrass, country rock, or grunge.

If you don't travel the Skyline Drive, you can make a stop for an hour at Natural Bridge just off I-81. DW and I spent a night of our honeymoon at Natural Bridge 37+ years ago.