Hi Pere Flyer,
I was just using the system yesterday and I think I can help.
I thought all stations had a ticket vending machine that takes cash or credit cards. But reading Mr. Dylan's, post, I could be wrong. Besides, if they are out of service - or if you get to the station just when the train is arriving, you could end up in a bit of trouble. That being said, according to Google street view, it looks like there are two machines in the tunnel right by the stairs to the platform.
Do you have a smartphone? If so, download the GoPass app for Android or iPhone. It will let you set up a credit card and purchase tickets that you then "activate" when you get on the train. You can even use it to buy tickets to the Texas State Fair and save $2. If you have the app, I'd get on at T&P and enjoy that station!
If you use the app, it's pretty simple, but there is one quirk that can drive you batty. It won't take "complicated" passwords. Gotta use just a 5-6 letter basic password. If it doesn't like what password you sign up with, it won't tell you why, it just acts like the app doesn't work.
That $10 regional day pass is a huge value and good for ANY The T, TRE, DART and their associated busses.
Wife and I were there over the weekend and took DART from the airport to Rowlett and back when we left. Loved it. Very slow compared to taking your own car, but one accident on the freeway, and we're zooming past, high above them.
Have lots of fun! If you have any questions, please let me know.