If you're goal is fat loss, not cardiovascular health, then lifting weights works just as well as running or using the elliptical, and it is easier on your body if you learn the correct form.
Strap an HR monitor on your chest when you do your next weight lifting routine, have it estimate your calories burned. Spend the same amount of time doing cardio at > 70% of your age predicted HR max. I think you'll find it's not even close.
Of course you said fat loss, and perhaps you'll quibble about how much of the calories burned comes from fat vs. muscle, but I think more people are trying to get to a healthy weight rather than achieve a single digit body fat percentage.
Sure - cardio is more efficient at burning calories than lifting... while you are exercising.
The point is that anaerobic exercise builds muscle, which boosts your basal metabolic rate. Unless you're doing cardio several hours per day, lifting will be much more efficient.
Even easier than that is just to not eat those calories in the first place.
With resistance training you are not trying to burn calories, you are building muscle. This new muscle will need calories to support itself and will make it easier to lose fat in the future because your body will need more calories to function.
> Strap an HR monitor on your chest when you do your next weight lifting routine
I've done that a few times. Consistently 140-160 BPM, peaking higher, sustained over 40-80 minutes.
My peak during Tabata sets is 180-190 BPM, so lifting isn't quite at that rate, but it's decidedly aerobic, especially for mid-high range sets (5-20 reps).
That's a big if. I don't trust myself to do it properly by myself based off a diagram from the internet. What's my alternative? The evil big box gym and overpriced personal trainer?
I hope someone builds a computer vision system to provide audio feedback for correct form. I often feel that I have zero self awareness for body position, and it makes me worried to work out alone.
Read starting strength. Its a small, cheap book, and it goes into more detail on form than you will probably need. It has chapters on each of the major lifts and I used to read it on my phone (kindle app) before/during the workout at the gym until I got everything right, using low weights of course.