do you push off with your left foot?
When you step out, you lift with that foot and then step out. You don't deliberately
push off with the opposite foot (there is weight transfer, yes, but to deliberately push off with the opposite foot means to transfer the weight onto that foot and then push off) when you move into court. That's one of the things you gotta unlearn.
or do you turn your body and step it out?
Exactly. And that's the thing. When you step out, you also have to turn your hip and torso a little bit in order to do this. It's not necessarily a side step (though it can be.)
Another rule of thumb -- though this mostly applies when moving into court -- is that you want your trunk to be "open" to the net at all times. This will accomplish the same thing. If you crossover, your torso will turn too much. If you push off with the opposite foot, you'll notice your torso will again turn away too much as you move to the ball.
Basically, you want to get to a point where you're comfortable tracking the ball with the outside foot, getting used to how the weight transfer and how the muscles should be used. Then you can incorporate all the other steps and more complex footwork patterns.
This fundamentally changes people's stroke production for most people. Because the weight transfer is usually not correct for most people when they pivot during the unit turn, they kinda lock out their hips. Hip rotation is not what most people think it is. What you see with the hips turning in a stroke is more indicative of the load on torso rotation. The actual
hip rotation is the weight transfer. So, when you execute the step-out wrong, you're actually no longer using your legs for power. Even if you push off or try sit and lift, you're only passively contributing to power. Moreover, this locking of the hips throws out your torso alignment. Most people are accustomed to treating their torso as a swivel-on-a-hip. Especially on the backhand side. But this is not correct (and unfortunately, the modern racquet and string technology is that you can get away with hitting this way without realizing that you're not loading properly.) The step out takes that away, so it may feel awkward at first. But you'll notice that you have (much) more weight into your shot, that you're better aligned, and that you no longer arm swings.