What Do You Hope to Accomplish?
The answer to this depends on what your goals are. Whatever you want to be better at you should do first. If you are a cross country runner then you should focus on running first and lifting some after because weights are’nt your main goal. If you are a bodybuilder and focusing on building muscle I would recommend you start with weights while you have all your strength and stamina and finish with cardio.
Most people can go to the gym for about 45 minutes to an hour. When you divide that time into weights and cardio you have to split the amount of energy you are able to produce to each activity. So instead of a good lift and a full on cardio session, it will be a quick lift and a quick run, slightly shortchanged.
When to Do Strength Training First
It takes a lot of energy to lift heavy and if you do cardio first, it uses up a lot of your glycogen needed for strength training and your technique might suffer due to fatigue. Unless your goal is endurance, I prefer doing weight training first. Even if you are trying to lose weight, strength training will help you see the quickest results. Resistance training builds muscle, and THEMORE MUSCLE YOU HAVE, THE MORE CALORIES YOU BURN AT REST!
Also another option If you are running short on time you can superset your weights with some cardio. You can do a set of curls and then do high knees or jump on the treadmill for a minute in between sets.
I like to start my workout by warming up for 3-5 minutes on the elliptical or row machine then I do my weight training. Whatever you do, its your choice. The fact that you are doing something triumphs doing nothing at all!