FYI, older popcorn may contain Asbestos.
I have removed the popcorn from the ceiling in my house and re textured it.
It is pretty easy; all you need to remove the old stuff is a pump style yard sprayer. You just fill it up with warm water and a little bit of TSP. Start in a corner and work in small areas (2’x2’). All you do is spray it down (not too much, not too little) then use a large putty knife to scrape the popcorn off (plastic is OK). I used a cardboard beer flat lined with a plastic bag, with a few holes cut in it for my fingers and thumb, then with the other hand I would use the scraper and just let the stuff drop into the box. You want to be careful not to be too aggressive and dig into the mud and tape underneath.
After it is all dry, you will probably have to go back and clean up the corners, and give the ceiling a very light sanding to knock down any goobers (with a fine sanding screen on a stick).
Next, you may have to get a large knife and some spackle to feather out the tape job to your liking. Usually ceilings that are popcorn textured have poor mud and tape jobs, because the contractors know that they don’t have to make it perfect when they are just going to spray that crap up there.
Next spraying new texture…
This is SUPER messy, mask everything…all walls, floor, EVERYTHING
Either call in a drywall contractor, or do it yourself. You can rent a texture gun for about $15. 00 a day, or just buy one for about $65. 00. The best thing to do is to practice on sheetrock scraps outside, adjusting the air pressure, the nozzle size and the thickness of the mud until you get a close match to your walls. Most homes have a spatter texture that was made with a pretty thin mud, a smaller orifice and higher air pressure. Then it is knocked down with a trowel after it starts to set a bit to make the surface flat. It is a soft science, and is something you need to practice yourself to get it right.
After the texture is dry, you give it a very light sanding to knock down any goobers, then primer it. Then you will most likely pull down the masking, and go about re-painting the whole room.