We get asked this all the time and our answer is the same: it depends on your goals, and here’s why:
If you are a regular gym goer with the aim of general keep fit, staying strong and beating the bulge then it is probable like 99% of other gym goers you will want to incorporate both lifting and cardio training into your workouts.
If you do a cardio session whether it’s a steady state or interval type session you will tire; you will have used up some energy stores. If in the same session you then want to perform some lifts, you may find your strength is depleted and/or your form is poor.
Alternatively, if you started your workout with a weight lifting session your muscles will be exhausted, having been trained to or close to failure and the chances of you pushing out any good cardio training next is unlikely.
So, the overriding answer to the conundrum depends on what you value as important in your own training and you may find it most beneficial to alternate sessions with one starting with cardio and the other weight lifting. In terms of science, recent studies have shown that the most cardio gain is to be had if placed before the strength training component of a mixed workout.
The same does not apply for those with less general goals: If you want to bulk then definitely put the strength before the cardio and perhaps consider interval/HIIT training over steady state cardio when it comes to it (but don’t forget it!). For long distance runners the cardio sessions need to take precedence however it is a good idea to have a couple of dedicated strength sessions pw to add to your long distance running as it is likely you will have run out of time or energy to add anything useful after a session.