Leaving a small amount of gas on when you push the clutch down to prepare for the shift can help smooth that transition as well. If you let off completely, you'll get one jerk from the throttle cut-off, another jerk from the clutch engagement, a third jerk when you let off the cltuch with the new gear, and a fourth jerk when you get back on the gas. That's a whole lotta jerking.
You'll prolong clutch life if you do your best to keep the engine torque fairly neutral to the car movement (neither slowing nor accelerating). So a little gas helps smooth the process because it helps the engine want to stay at the proper revs by itself, so the clutch doesn't have to do the work of spinning it up or slowing it down. After you are fully off the clutch, then you can start to feed in more throttle and let the car accelerate. But you shouldn't be doing any hard acceleration while slipping the clutch except for the first few feet of first gear.
The only exception is when you want maximum speed at all costs. Then you leave the throttle pinned to the floor the entire time, dumping the clutch on each shift and letting the clutch yank the revs down each time, which is brutal on the entire drivetrain but gives you an extra oomph on each shift. Mazda encouraged car mags to shift this way in the Mazdaspeed 3 during testing to avoid losing turbo boost between shifts and make their stats look really good. But you would never want to do this to your own car.