Take Highway 7 north to the junction with Hwy. 10 at Ola. Go west on Hwy. 10 to Boonville where you'll take Hwt. 23 north to Eureka Springs. This route is a couple of hundred miles and around six hours. Also this is the scenic route. If you're looking for the quickest route take Hwy. 7 to I-40 west to I-540 north to U.S. Hwy. 62 east to Eureka Springs. This is a longer route but will take less time due to a lot of it on interstate highways.

I second the advice to take Arkansas Scenic 7 and Arkansas 23 (aka The Pig Trail), if you can spare the time. This is a great route. 23 will take you thru the middle of the Boston Mountains. For some more information about the routes: www.arkansas.com/things-to-do/scenic-drives

Thank you so much for the advice! Now, Part 2-

could you please recommend a quicker route to travel from Eureka to Memphis?

Also- from my past traveling experience- including the Rockies and northern Spain- is the "scenic" trip from Hot Springs to Eureka Springs a nail-biter or simply slower?

Arkansas Hwy 23 is a two lane road. The last time I was on it it was in good repair. It has a number of sharp turns but I wouldn't call it a nail biter.

Here's an alternative route from Eureka to Memphis

From Eureka take US Hwy 62 east to junction with US Hwy 65 South

Stay on 65 to Conway then get on US 64 east

At Beebe 64 will merge with US 67/167 north

At Bald Knob take 64 east to I-55 south

I-55 to I-40 south into Memphis

Here's another possibility

From Eureka Springs take US Hwy 62 east

At Imboden take US Hwy 63 south

Stay on 63 to the junction with I-55 south

Take I-55 to junction with I-40

Stay on I-40 into Memphis

If I were you I'd get on Google Maps and check out these routes for myself.

Even tho Hwy 23 will take you directly south, you might still want to consider taking it just so you can hit I-40 faster.

Thank you so much for your helpful advice! The only concern now is that the price of gas doesn't double by summer! Again, thank you for your help.