I want one command block to activate on the the first pulse but another when I hit the button a second time.

You need a flip-flop (more specifically, a T flip-flop). In electronics, a flip-flop is a circuit that toggles between two states. Here is how you might build a T Flip Flop in redstone:

Front

Back

I borrowed the design from the Gamepedia's page on Minecraft memory circuits. How does it work? Press the button, and the output turns on. Press it again, and the output turns off. (The output is the trail of redstone at the bottom right of the first picture. In both pictures, output is "On".)

Note that a wooden button will not work with this design, but you can find other designs on the linked page that will work with wooden buttons.

That's cool and all, but how does this solve my problem with the Command Blocks?

Take the output redstone and split it into two trails. Invert one of the trails. Now one is always on, and the other is always off. Problem solved.

Below is a picture of how to do this. Pretend that the top left torch is the output of the first circuit.

I want one command block to activate on the the first pulse but another when I hit the button a second time.

You need a flip-flop (more specifically, a T flip-flop). In electronics, a flip-flop is a circuit that toggles between two states. Here is how you might build a T Flip Flop in redstone:

Front

Back

I borrowed the design from the Gamepedia's page on Minecraft memory circuits. How does it work? Press the button, and the output turns on. Press it again, and the output turns off. (The output is the trail of redstone at the bottom right of the first picture. In both pictures, output is "On".)

Note that a wooden button will not work with this design, but you can find other designs on the linked page that will work with wooden buttons.

That's cool and all, but how does this solve my problem with the Command Blocks?

Take the output redstone and split it into two trails. Invert one of the trails. Now one is always on, and the other is always off. Problem solved.

Below is a picture of how to do this. Pretend that the top left torch is the output of the first circuit.