My First OverHead Projector

I've had good luck with a couple different Epsons and JVC models, all 3k and below when I purchased them. I use them on a 110" screen in an 18x 30 room. A lot of specs are tricky because there's no true standard, so depend on reviews. Projectorcentral has good info.

Look at the specs and get the best specs you can afford. Unlike audio equipment where specs don’t always mean musicality, projector specs DO determine the image quality. As mentioned by others, I would definitely stick with new!

I have a BenQ 3550, a nice 4K pixel-shifting projector from about 2020. The TK860i is a close replacement for under $2,000,  and the somewhat more expensive 4550i is an LED unit that kinda owns the $3000 price range right now. My next buy. 

The Epson 5000 series were notorious for bad power supplies (I went through 3 before calling it quits). Panasonic has always provided good value and reliability, if not class-leading performance. 

Two things to consider: bulb type projectors put out about as much heat as a room heater, if your room isn't well ventilated, you may get pretty warm. You can spend a ton of money on a screen, or make a perfectly good screen by painting a piece of 5X9' MDF trimmed to 9X16 ratio  (google 'painted projector screen' for paint mix) and hanging it from a French Cleat for about $200. You can also get 'you assemble' kits in larger sizes in the $500 range. 

Screens are a mystery, what with gain, ambient light rejection, etc. If your room is dark, you likely won't need ALR, but if it has significant ambient light, the an ALR screen helps, but note that they can hot- spot on axis. 

At 110" diagonal in my theater, I can tell you that no direct view TV even comes close. And for sure, as the screen gets bigger, the difference in 4K becomes very apparent. 1080 material will always look kinda soft and off-focus. with a 4K projector though, your AVR should upsample to a good facsimile of 4K.