Would a Dallas Personal Trainer say to jog outside in the sunshine or inside on the treadmill?

Well, it depends on which Dallas personal trainer you ask.  If you ask a trainer who's interested in genuine health you will probably get a response that favors the outdoors, if you ask a trainer who is only interested in the physical appearance, they may favor the indoors or have no preference whatsoever.

There are certainly cases where it makes more sense to use the treadmill such as when you are trying to exercise at a certain specific speed...especially if you are fluctuating between differing speeds during the same cardio workout.

Outdoor running in the sunshine will give you the opportunity to soak up some healthy immune system boosting Vitamin D, fresh air, and will give your legs the more natural motion of running on an actual surface that must be pushed off of rather than assisted by the treadmill's belt.  More on this treadmill