There are basically two questions that come up: first, do I think it’s a good idea to do them on the same day; and second, which one do I do first?
The simple answer to the first question is, yes, it’s fine to do them on the same day. However, it’s a matter of priorities. This brings us to the second issue–what to do first?
Think of it this way; after you run 5 miles, you probably won’t have as much energy to devote to your strength training. Conversely, if you lift weights for 45 minutes first, your cardio session is not going to be as productive.
Life is like that though, there is no one best way to do anything. All situations have pros and cons.
This is why I typically like to alternate my strength training and cardio workouts. (Although that system isn’t perfect either!)
If you want to experience the best of both worlds, another possibility is to combine strength and conditioning at the same time by doing circuit training or kettlebell workouts. Just don’t get limited by only doing kettlebells or circuit training. They’re great ways to maximize your time, but they’re not the be-all-and-end-all of fitness.
If you are training for improved performance in a given activity, be it a one rep max on a deadlift, doing 50 pull-ups, or running a race, you need to train specifically towards that goal and make it a priority. However, if your goals are more loosely defined, then feel free to experiment with different approaches to see what works best for you.
Like I always say, there is no better way to learn than through your own firsthand experience.