How do I know if I need a special primer before painting basement?

UGLI latex. It waterproofs as well as works as a primer. Get it at Home Depot.

https://www.familyhandyman.com/painting/painting-how-to-choose-and-use-primers/view-all/

https://www.kilz.com/concrete-coatings/kilz-basement-masonry-waterproofer

We painted our cement block with a paint made for basements amd cinder block. It was called basement and masonry waterproofer, and can be tinted to many different colors.

use a moisture block paint made for concrete/masonry/brick --- and be sure the wall is bone dry! these paints can be tinted to you choice of colors

Ask your paint dept at local big box store for the best concrete paints and sealers. If you paint the ceiling tile it will take away some of the sound proofing elements but it will clean up the area.

I would also use Drylok to prevent moisture. Yes you can paint ceiling tiles, those I would prime as they are usually very porous. If they are the acoustic tiles with the little holes, they make a foam roller with 1/2" cuts in it for that purpose, I got mine at True Value and they have them on Amazon too.

Special primer is needed if room's got high humidity and mold. Also you need heavy-duty primer to cover dark stains.