Strength training your legs does increase punching power.
All of the following assumes you have access to some kind of strength training equipment, preferably barbells and heavy ass weights.
You're going to want to do:
1) Squats. I'd suggest doing front squats with a proper clean grip over back squats, but do whatever is more comfortable. The two different kinds of squats work different muscles, but it's not that big of a difference that you should sweat, especially if you've never trained legs/squatted at all.
2) Deadlifts. Pull conventional, or use sumo stance, whichever feels better for you. I'd stay away from stiff legged deadlifts and good mornings for the time being.
3) Lunges. I prefer walking lunges because they seem to have a better "stretch" than other kinds of lunges, but do lunges however you want. You don't really have to do them, but they're a really good strength training tool that I like to incorporate for a number of reasons.
Squats and deadlifts you'll want to keep reps per set low, with heavy weights. For example, you might do 6-10 sets of 3 reps. For lunges I like to go medium weight, medium reps. For example, 3-4 sets of 20 reps (10 reps per leg).
A lot of power comes from your core, and the fact is is that when you work out your lower body through heavy compound movements, you work your core as well. This is because your core needs to be strong and brace itself, otherwise the weight will pull your torso over and dangerously round your back out.
Power in a good punch starts at ground level. You almost "push off" the ground with your legs to start the development of power, and it travels up through the rest of the body from there.