If you were an avid hang glider pilot at the time you posted your question, you no doubt have long since connected into the local hang gliding community and found the answers to your questions, and have spent many joyful hours soaring under the beautiful cumulus clouds over the Texas flatlands.
Aerotow (towing up behind a slow-flying ultralight or lightsport aircraft) is one way to launch hang gliders from flat ground. A typical tow will be to around 2500' or 3000' up. You'll start with the glider on a three-wheeled dolly so your own legs don't have to be involved. If you've never done it before you'll get training in a "tandem" hang glider, flying with an instructor. There is an active hang gliding business very near Houston that offers aerotow launches, and I hope you have made contact with them since you posted your question. (Google "Cowboy Up Hang Gliding".)
Another method is called "platform launch". The glider is placed on a platform on the bed of a pick-up truck. The truck has a reel of line that pays out under constant pressure, regulated by a disk brake. The truck starts driving down a long little-used road or on a dry lakebed or an airport runway. When sufficient airspeed is reached to ensure a strong positive launch with a strong vertical climb rate, the pilot pulls a trigger to release the glider from the truck and the line starts spooling out. I've gained as much as 4000' of altitude during this type of launch, using a reel of lightweight spectra line, and a 3-mile-long road.
This type of launch is a very technical procedure and unless you are a willing to take high risks as a test pilot, you must use the exact equipment that other experienced flyers have carefully developed over the years. You must get training in the techniques, use a detailed checklist, etc. I'm not sure if this is currently being done anywhere near you. Aerotow would be the easier route in most cases.
Yet another method uses a stationary winch that reels the line in. Often such a winch is made from a motor scooter, mounted on a trailer, with the back wheel replaced by a drum (reel). This method can be good for beginner training via ground-skimming flights, but also can be used to launch to several thousand feet of altitude.
It is also possible to tow a hang glider from a car or truck using a fixed length of rope. This is not as simple as it sounds-- for example a line tension gauge is essential. Read on for more safety tips.
To safely tow a hang glider, you must avail yourself of community knowledge and training. With many methods, it is vital to have some way to actively control the towline tension, and with all methods, a weak link of appropriate strength is essential. Reliable release mechanisms that route the towline's pull to the appropriate points on the pilot's harness (and sometimes to a point on the glider as well) are also essential. Simply tying one end of a strong rope to the glider and the other end to a fixed point on the ground and trying to launch yourself like a kite on a windy day would be extremely dangerous. Several fatalities occurred this way back in the early days of hang gliding, before the community gained expertise in safer methods, and also developed a strong culture of safety-consciousness. A phenomenon called a "lockout" is waiting to bite you if things go wrong. If you've ever seen a tethered kite dive into the ground at high speed with the string under high tension, you'll have some idea of what this might be.
Generally speaking, hang gliders have less control authority than sailplanes, especially in the roll axis, so towing hang gliders is more complicated than towing sailplanes, and more care is needed to maximize the odds of a safe outcome. However, if due caution is taken, towing can be a very effective way to get a hang glider up into the sky.
There are also motor units available for hang gliders. The modern method is to incorporate the motor into the pilot's harness, not to attach it to the glider itself. Using your own feet as landing gear while launching with a motor can be somewhat challenging and also dilutes the joy of pure soaring flight; you'll probably have a more enjoyable experience if you go the aerotow route. But google "mosquito harness" or "hang glider powered harness" for more on the motor option.
A related concept is the "light trike", where there are wheels and a seat as well as a motor, all connected to a hang glider wing, but everything is kept light enough in relation to the wing size to allow a low sink rate in unpowered flight. The motor is used to get some altitude, and then you shut it off and have a nice soaring flight. The control "feel" may be a little awkward compared to a pure hang glider flown from a prone position-- most hang glider pilots would prefer to let the motor fly away with the towplane at the end of the tow, as long as the soaring conditions are good and a towplane is available.
I can't answer a question about hang gliding in Texas without referencing the famous flights originating from Zapata TX on July 8 2012-- one pilot flew 764 km (475 miles), and a second pilot flew just a few miles short of that distance. Read more about it -- and see video of their aerotow launches-- here: https://xcmag.com/news/dustin-martin-and-jonny-durand-break-hang-gliding-world-record-in-zapata-4/
And yes of course it is also possible to run into the wind as fast as you can and then increase the angle of attack of the wing to do a little hop. In terms of peak altitude gain, this is particularly effective on a beach, where the wind gradient tends to be especially pronounced-- the wind gradient will help you out as you start to climb-- but then will work against you as you come back down. You'll never get more than a few seconds of flight time this way, but it can be a fun thing to do a few times after you touch down on the beach after a nice long ridge-soaring or dune-soaring flight on a windy day.