While it's true that building more muscle will raise your metabolism, weight lifting by itself will not add muscle. Weight lifting using high weight and low reps combined with a high calorie, high protein diet is how you gain muscle. For the average person, gaining muscle and losing fat at the same time is not possible cause one requires eating more calories than you normally need and the other requires cutting back on calories.
However if your goal is fat loss, you SHOULD still weight train. There are two reasons for this. One, when you go on a diet, you risk losing muscle along with fat. When the body is getting fewer calories than it needs, it reacts by getting rid of the most expensive tissue to maintain which is muscle. By doing weight training, you send a signal to your body that you need that muscle and that it shouldn't get rid of it. Second, when if you use low weight and high reps, meaning a weight you can do for 10-12 reps, 4 sets and 90 seconds rest in between sets, you start building up lactic acid in your muscles. Lactic acid causes growth hormone to be released and one of the things growth hormone does is move stored fat into the bloodstream. This is also the mechanism cardio uses to burn fat and why people burn fat when they're sleeping, cause that's also when the body releases growth hormone.
What I would recommend is alternating your workouts. Do weight training 3 times a week, every other day. Do cardio on the other days and give yourself one day to rest. Do the cardio first thing in the morning on an empty stomach and wait 30-60 minutes after you're done before eating breakfast. Also, as soon as you're done weight training, drink something with a lot of simple carbs, like gatorade or a soda. This will replenish your muscle glycogen and prevent your muscles from breaking down.
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Cardio, Weight lifting, and Weight loss 
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