Cool is the singular currency of Paradox. You spend Cool to make your Agent act with awesome ability, and you earn Cool by both rolling well and playing your Agent with gusto.
The limitation of Cool is that is can't be kept. Cool lasts only for a short time, one instance in the rules. Once the action dies down and the dice stop rolling, You have to use Cool or lose it.
There are two ways around this limitation. First, you can Press to continue the action and carry over Cool (-1) to the next instance in the rules. Second, if you have five Cool, you can buy 1 Awesome.
Awesome is outside the rules almost entirely. The only function of Awesome is to allow you to buy Cool when needed. You can burn one Awesome for 4 cool all at once, or tap it for less. When you tap Awesome for less, its worth 3 the first time then 2 the second and 1 the third. Once you've tapped it, you can't burn it every again. You can note tapped Awesome by placing a -1 and -2 next to it, meaning you will get 3 - 1 = 2, or 3 - 2 = 1 Cool the next time you tap it. Once you have tapped it three times, its used up.
In order to Press the action forward and carry your Cool over with a -1 cost, your Agent must either: Escalate the current conflict, or Advance to the next logic conflict in the fiction. You need to consider the term conflict here as a broadly scoped thing. A conflict might be: Agent Smith wants to destroy Mr. Anderson. Which means all kinds of smaller contests can fall under it. You should note that either method may not occur immediately following the current actions in the fiction, but must logically follow the current actions.