I'm closer to Dallas so these recommendations will probably skew towards that side of the metroplex. It's Texas so it's a given you need to have some BBQ; my personal favorite is Pecan Lodge given their high quality and consistency. Second behind that is Cattleack which is also excellent, though I'm not as wild about their ribs. I have not yet been to the Terry Black's, but have heard great things about it. Additionally you'll hear about tons of other places such as Lockhart, Hutchins, Slow Bone, Heim, Goldee's, etc which are good in their own right, but if I could only go to one it would be Pecan Lodge. Both Terry Black's and Pecan Lodge are downtown in Deep Ellum which is a funky/artsy area (though a bit dirty and occasionally sketchy at night).
If you want steak there are a few that always pop up as recommended--Nick & Sams, Bob's Chophouse, Al Biernat's, and my personal favorite of Pappas Bros; the dry aged ribeye will melt in your mouth. Tacos I mainly get up near me about 30 minutes to the north so I'm not as well versed in the downtown offerings; for a first timer it's hard to beat Fuel City to truly understand a great taco can come from anywhere (don't skip on the elotes cart). Burgers I'll usually go to Maple & Motor or Keller's Drive in, but there are a handful of decent places around.
We don't have as much as far as trails goes; with the sprawling metroplex the closest offroad park is called Lonestar Offroad Park, but it's gotten a reputation of essentially being a giant mud pit. To the northwest of DFW you've got Northwest OHV Park in Bridgeport and to the east by a bit (2hrs) you've got Barnwell Mountain which is home to Toyota Jamboree.
If you have some spare time downtown the Sixth Floor Museum at Dealey Plaza is fantastic, as is the Perot Museum. Downtown will have a variety of art installations spread out; Klyde Warren Park is a fun hangout and they have a variety of events If you make it to Fort Worth it's always fun to go see the cattle drive and walk around the Stockyards. Renowned Texas chef Tim Love has a handful of restaurants all over that side of town and a neat one is Lonesome Dove; they have a relatively exotic menu where you can try a number of game animals. The Dallas Arboretum is beautiful, but in late July/early August it will be very warm out there; it's near White Rock Lake which has a pretty incredible 9.5 mile loop around the entire lake (though again, not as enjoyable in the summer).
There's plenty more, but that should at least get you started on looking into some stuff to do