I have actually used the public transportation option before.
You take the #39 bus from Love Field (I believe that's the only route available from Love now) downtown where it eventually brings you to the West Transfer Center (a route map is available on DART.org).
From the West Transfer Center, it's a very short walk over to the West End (light rail) station, from which you can ride the fairly short distance to Union Station.
From Union Station catch the Trinity Rail Express which runs to Fort Worth, but get off at CentrePort/DFW station. There are shuttles which run between there and DFW airport.
You can buy an All Day Pass when you get on the bus that also covers the Light Rail and the Trinity Express--but because DFW is slightly over a Fare Boundary, the required pass is the "Premium" ($4.50) all-day pass rather than the "Local" ($2.50) all-day pass. Quite a good deal, if you have a LOT of time to get between airports. It's helpful to check the bus and Trinity Rail timetables ahead of time (at DART.org) if doing this--the light rail service is pretty frequent, and either the Red or Blue line goes to Union Station.
At some point in the past there was another bus route running out of Love which stopped at Medical/Market Center station, from which you could also catch the Trinity Rail but without going all the way downtown first. Don't remember the route # but am pretty sure it doesn't go to Love anymore, unfortunately.