We went through this process several years ago and had the popcorn removed from about 900 sq. ft. of our home.
We found a fellow from Chehalis, WA to do the job. At first there was a concern over asbestos since our house was built in 1972, but knowing the contractor and his quality of work it was determined to be safe.
The process started by moving all the furniture to the center of each room, then covering the walls, floors and furniture with plastic. He used a special steamer to loosen the popcorn texture before scraping. This also protected the sheet rock from any damage (and also set off our smoke alarm at one point).
After the ceiling was cleared, the sheet plastic was bundled and removed. Once everything was dry, the old ceiling drywall was secured in spots from any sagging and repairs were made where necessary.
The final step included more clear plastic sheet for sealing the sheet rock, texturing and painting. In addition to the ceiling work he worked with me to remove some cedar paneling from one wall, then seal and texture it. We painted the wall and replaced the baseboard trim when he was done with his work.
I'm glad we contracted out the work, but we were lucky to get someone experienced and local. It's a lot of back breaking work. He did the job with the help of his wife and daughter.