run then weights, or weights then run, and why?

Weights first then cardio. When I do weights first I use all the easy carbs and sugar, then running burns more fat at that point because my body tries to conserve the last of the easy carbs and sugar.

The trainers at my gym taught me to do a short warm-up of light cardio, weights and then at least 30 minutes of cardio. The reason is that it takes your body 30 minutes to burn your glycogen stores that you have and you can't start burning fat until those stores are gone. So, by starting with weights you are buidling muscle while burning your glycogen stores and then your cardio is full-on calorie/fat burn. That's what I do and it seems to be working fairly well. Others might have different stories.

I've always done my run first and then my weight training, it gets my heart rate up during the run and then keeps it up during the weights. For example, tomorrow I'm returning to the weight room (I've only been lifting in group classes for the past 3 years) and was planning on getting there 30 minutes early to run before I met up with my PT. Which order to do you it in and why?

I've always done exactly what you do for exactly the same reason. I do see a lot of the guys, who don't necessarily like cardio, doing laps or sprints around the track during their weight training. My gym's PT says that those short bursts of getting your heart rate up and then back down combined with your weight training is the ticket. But, I can't go without my daily run, so I don't do that.

I do a 10-15 minute warm-up and then weights and then do my 40-60 minute cardio after. My reason is it takes the body 20 minutes of working out before it starts to burn fat. So if I warmup and then do weights warming up all the muscles by the time I hit my cardio full on I am just a fat burning machine. This was set up by my trainer doesn't mean it's the same with everyone. This site has a lot of different opinions, so do what is best for you.