If you've never been to Texas the map might surprise you. I met an Ohio fellow a number of years ago who was a factory rep and making calls in Dallas, Houston, and San Antonio. Being accustomed to the distances represented on an Ohio map, he thought he could make all three calls in one day, and was hard pressed to accept that we weren't pulling his leg about how long it would take him to drive between those cities.
El Paso is far west Texas, Longview is near the eastern border. When in El Paso you are closer to San Diego than to Longview. If you folded Texas to the west at El Paso, Longview would be in the Pacific ocean. If you folded Texas east at Longview, El Paso would be in the Atlantic ocean.
The point is, with six days with which to work you'll have to prioritize and pick, the distances are too great to do even 20% of the things listed in the above posts.
From a Denton starting point, you might do something like head west on I20 to Highway 16 and turn north 4 miles to Mary's Cafe for lunch. Then head south on 16 for Fredericksburg, (German food) Kerrville, and west to Medina for the east end of RR (Ranch Road) 337, the best of the 3 sisters (ranch roads 335, 336, and 337). About a day and a half of non-stop riding will pretty well cover that part of the Hill Country, then make your way to San Antonio for the River Walk and Alamo. Then Luling and Lockhart for BBQ, and back northwest to Hwy. 281 north. I would avoid Austin simply due to the traffic, which is a shame because it's a great party and music town, and the State Capitol building is an interesting stop. Stop in Marble Falls at the Bluebonnet Cafe for homecooking or if between meals at least a piece of pie. Make your way back to Fort Worth to the Stockyards, Cattleman's, and maybe Joe T. Garcia's (just south of the Stockyards) or La Familia (just west of downtown and south of west 7th street) for Mexican food. East to Arlington to see Ranger and Cowboy Stadiums. Back to Denton hopefully with time to ride 455 from Hwy. 75 west to Montague and back (going through Tioga there's pretty good BBQ at Clark's Outpost on 377). Or out of Montague you might want to take 59 to Saint Jo, 677 north to 373 and ride back through the wind generator fields to Muenster (more German food).
Something like that will pretty well take up your 6 days, and you won't have gotten within hundreds of miles of other great Texas destinations like Palo Duro (ride 207 if you choose to go to the Panhandle), the coast (Padre Island, Corpus Christi where you can tour the Lexington and the State Aquarium, and Big Bend and the River Road (which would take up your 6 days, and it's absolutely the wrong time of year to be there due to the heat).
This got a lot longer than I intended, I hope it provided some food for thought. Enjoy the great State of Texas and please give us a ride report after the trip.