No, they don't cancel either other out. Most athletes do some combination of weights and cardio. I think whoever told you that is confusing cutting and bulking, which do cancel each other out. Think of cardio as lifting for your heart and lungs.

I suggest you do more leg work. Squats burn mega calories!

Personally, I think this idea of losing all the bodyfat before you start to build muscle is dangerous. You end up very skinny, and find it very difficult to build muscle afterwards. If you lift, you'll build a good muscle base which will help you lose the fat when you need to.

But the best way to get rid of fat is diet. It's much easier to cut out a bagel from your daily diet than to exercise away a bagel's worth of calories. Exercise alone rarely works, because you get hungrier as as result of the exercise.