In the bad old days there was no way to get to the DFW airport except a good buddy with a car, an expensive shuttle or even more costly cab.
Dallas has come a long way with public transportation since the 70 and 80s!!! There are a few transfers involved, but you can get from the big airport to Garland/Richardson using a combination of the Trinity Rail Express (TRE) which is a sort of commuter train. That takes you to Dallas Downtown Union station. From there a transfer can be made to the DART Light Rail. The Blue,Red, and Orange lines going North will take you into Garland (Blue), Plano and Richardson (Red and Orange.)
The hardest part of all this is actually getting to the TRE! At the airport, go to the lower level baggage claim, then go outside to get the free shuttle bus going to the Remote South stop. From there, there is a connecting bus a few feet away to TRE. It is a little inefficient, but works well in the end.
Beware, as of the date of this post the Saturday service is sparse on the TRE and DART. NO SUNDAY SERVICE ON TRE!!!!
So, it is not a 7 day a week solution.