Armadillo's or Kevlar Inserts? So, I've had at least two flat tires on my way to work. Therefore, I'm tempted to splurge on  a pair of Specialized Armadillo's (not the elites...)

However, I am not a particularly skilled rider.  I had a pair of racing slicks from the late 1990's (at the latest!) and they were a tad on the nightmarish side - I slid out at 5 miles an hour trying to leave a parking lot.

Seriously - slicks were enough to give me an existential crisis. 

At the same time, I'm notorious for being, err, close, yea, that's it.  Close to late.  As such, I don't really like flat tires.  My choices at the bike shop came down to this: cheap gumwall tire with treads and Kevlar Inserts, or a pair of Armidillo's which have minimal treads.

Which should I choose? I commute in all weather conditions.

Speed is nice, but I really care more about not dying and not calling in late from the side of the road than minimizing rolling resistance.
Your sliding out at 5mph wasn't because the tires were slicks, it was for some other reason. They were either lousy tires, improperly installed, rider error, or some other factor.

If you are riding on the road, there is no need for any kind of tread. Slicks are perfectly fine, despite your thoughts from your previous experience. In fact, they are ideal, and will be easier to handle on the road than tires with tread. A definitely don't get any of those combination tread tires, those are the worst.

Forget the treaded tires with kevlar inserts. Get the Armadillos, Continental Gatorskins, or some other sort of flat resistant tire. Make sure they are always pumped up before you ride, and they'll serve you well.