Our story begins by me describing how much I do not like getting up.

To get up means to get out of your bed, to stand up after you have been lying down on a bed.

I say that, "The worst part of my day is when I have to get up.

Waking up, that I can handle?" To wake up, "wake up," (two words) means that you are sleeping and you stop sleeping and now you are awake.

The verb is to wake up. So, you can wake up while you are in bed, and then, you get up--you stand up after lying down.

I said that "waking up" is something "I can handle" To handle, "handle," here means to be able to control--something that I can manage--something that I can accept; it's not a problem.

Another expression would be something I can deal with.

To deal, "deal," with something is the same, in this case, as to be able to handle something.