A few things come to mind there:
1) The draw. I've seen old videos of me and it's embarrassing. Tons of wasted motion. Up with the gun, then down low, swooping back up toward the target, arms going all over the place. We dubbed it the "potato-digger draw." Learn the 4-count draw stroke and practice it until it's all you know.
2) Reloads and other manipulations. Watch some good ones. Practice these minimal-motion steps, slowly at first, until you don't waste time fumbling around. And for heaven's sake, learn to work up in front of you. In your "workspace." Don't try to reload and/or clear jams down at waist height, bent over like you're trying to see your shoes. Do this so you can work quickly, keep your balance, keep your eyes on the threats/targets, and push straight back out when you're done.
3) Use of cover. IDPA has rules for using cover. They do NOT require you to hug walls, tiptoe up to edges, or bob back and forth between shots or to reload. (Check out our "Don't Crowd Cover" thread for more info.) This is a huge time waster for new shooters. They move like kindergarteners playing hide-and-seek.
4) To that same end, once you are able to run the gun without conscious thought, start to pay attention to the choreography of what you're trying to do. Many shooters waste a lot of steps, and a lot of time, taking lines that are unnecessarily tight or going places in the stage they don't need to go. What is the actual shortest distance between where you intend to shoot? Where do your feet need to be to keep you in cover? Can you keep in mind where a target was beyond a barricade so that your gun is already presented and trigger staged as you come around the shadow of cover? If you don't have to go hunting for the targets, you'll shoot a lot faster.
4a) Shoot as slowly as you have to to get hits, but when you're not shooting,
HUSTLE! None of the fast shooters I've shot with trot or amble between firing points. They're at a sprint if the gun isn't firing. All that time in movement? Yeah, the timer's counting that, too, and on some stages you could lose as much time there as in your points down!
5) Lead with your eyes. Get your eyes and head moving ahead of the gun so you're already "on target" mentally as the gun catches up. This is more advanced, but your transitions will benefit greatly.
6) Forget the double-tap. Every shot is one shot. You're just shooting them as fast as you can see the sights. And as 9mmepiphany can tell you, you can see those sights MIGHTY fast! Combine this with number 5, and you'll be amazed at what you can do. Eventually you'll run an "El Prez" drill that sounds not like "bu-bang...bu-bang...bu-bang" but as an even string of shots "bang,bang,bang,bang,bang,bang" in cadence, and they'll be Down-0, too!