We did three different things to cover up our popcorn ceilings in three different rooms.
In our downstairs room, we tore down the existing ceiling and put new drywall up (got to put in new lights and a ceiling fan while everything was exposed too!).
In our upstairs, we have loose fiberglass insulation in the attic, so we cannot tear down the existing popcorn ceiling without making a huge mess.
In our upstairs master, we scraped off the existing popcorn using a spray bottle to soak the popcorn, and a drywall mud applier-thing as a scraper. We were renovating the room, so we just let everything fall on the floor and cleaned it up afterward. The scraper gouged big chunks in the ceiling that we had to repair. It looks ok but not as nice as I would like. It took a long time ~4-5 hours for 200 square feet, but it was really tiring since everything was over my head and my arms got tired really fast.
The other upstairs room was a hybrid. I scraped off the popcorn very roughly, not caring if I scraped the heck out of the underlying ceiling, in order to make a relatively flat surface for new drywall. It took about an hour or two for 150 sq feet using my spray bottle and scraper again. Then I put new drywall up overtop the existing, now very scraped up, ceiling. I adjusted the ceiling bracket that holds the ceiling fan (the only thing hanging on the ceiling) so that it was flush with the new ceiling (used 1/2 inch drywall and 2 1/2" drywall screws). It looks as good as the new downstairs ceiling.
Disclaimer here, my neighbor is a professional dry-wall hanger, so when I say that I hung the new ceiling, I mean we (me, him and his partner) hung the new ceiling. We help each other out, so it was a free project for me minus $200 for another one of his buddies to do all the finishing (mud/sand, etc) on a Sunday afternoon and another night after his normal work hours. Without free supplies and labor, I probably would have just scraped the popcorn off all three ceilings and never had bothered putting in the new lights and ceiling fans.
Like everything in life there is a spectrum of quality versus cost. The cheapest route (scraping) looks like the ceiling was scraped (but still a lot better than before), the most expensive route looks like new construction (not shoddy new construction, proper new construction :) )