The question always arises whether weight lifting (or resistance based training) should be performed before or after aerobic exercise. Before you decide to take the plunge on one or the other you should consider your overall goal; whether it be weight loss, improving lean mass, or a bit of both.
During a single bout of aerobic exercise the body utilizes energy stores consisting of fat, glucose stores, and circulating glucose (blood sugar) for fuel. Some fuel sources will face an imbalance following exercise depending on the intensity/duration, which will require a rest period and/or dietary replenishment to recover the stores. If your goal is weight loss, greater emphasis should be placed in the aerobic exercise category while continuing to engage in resistance training for overall health benefits. Studies have shown that aerobic exercise encourages greater excessive post-exercise oxygen consumption (EPOC); meaning your body burns excess calories after exercise has ceased therefore promoting weight loss. On the other hand, resistance training in larger volumes (ie: more sets and less repetitions) promotes muscular hypertrophy and lean muscle building. So, if your goal is to improve muscle mass, your focus should be primarily voluminous weight lifting with some aerobic activity for overall health benefits. Finally, if your goal is to lose weight and improve lean muscle it may be advantageous to have a balanced program that focuses on either each of the exercise types in an alternation or that involves mixing the two together.
Overall, whether you decide to resistance train or exercise aerobically first, just know your performance may suffer in the latter. For example, if you were to weight lift prior to jogging/running, you will face muscular fatigue and poor performance due to your energy stores being somewhat depleted. If you were to run prior to lifting weights, your lifting form may suffer putting you at risk for injury as a result of muscular fatigue. Unfortunately, it may take some time for trial and error before you find what works better for you. Just keep in mind your overall goals and how you can adjust your training plan accordingly with the information provided above.
Happy and healthy training!