How do you get faster...

This video helped me out.  Then you just need to do it a couple of times.  Good luck.

Changing a tire

...at changing a tire? Seriously. It took my like, 25 minutes to fix a flat this morning. I suck!

You get faster by practicing. A lot. 

 It was also the rear tire, and I cannot tell you how much I struggle with the stupid gears when I'm done fixing the flat!

It helps me immensely to shift to the smallest cog before removing the wheel.

So, some questions, though I'm gonna go bug my LBS tomorrow and see what the heck I'm doing so wrong that it's taking almost a half hour to fix a flat. 1. Do you take the entire tire off the rim, or just one side so you can pull out the tube?

Just one side. Pull out the tube and search for the cause of the flat before putting in a new tube.

 2. And also, sort of unrelated, how do you know when you've used the entire CO2 cartridge? I thought the tire was completely inflated, but a little was left over (and escaped out of the pump when I pulled it off the nozzle). Is that bad?

You should feel the gas stop moving in, whne the pressures equalize. Yeah, they'll be a little left but it doesn't matter.

2a. Does the cartridge inflate the tire to **exactly** 120 psi? So many questions. And no time whatsoever. I swear, if this ever happens to me during a race, I'm so screwed.

Depends on the size cartridge, I think a 16g cartridge will get you close to 120 whereas a 12g is more like 90psi.

OK, brb, gonna go sacrifice a chicken to the tire gods. Sigh.

I prefer steak.

If you don't already have a trainer get one for this winter ( I see your from NJ). 

Use the same rim but use one tire (an old one) for the trainer and another for your outdoor rides.  Every time you switch them you are effectively changing a flat (sans the CO2)

Onerous, heck yeah,  but after one winter  you'll be able to change a rear flat in no time. 

Kill 2 birds... improve your bike and flat changing skills.