Shower and sink drains fill up with body hair. The lye products (Drano, etc) break down the hair. The trick is use plenty of cleaner, and give it plenty of time. Then rinse with hot water.
If you have kids (or previous occupant), there may be plastic toys in the drain, you never know.
Sometimes a slow drain is caused by a clogged vent... you know, those little pipes on the roof? Get up on the roof, find the vent over your bathroom, and run a snake down it.
You may have to cut into the wall behind the tub drain to access the trap, remove it, and snake from there. Sorry, it may have to be done. Cut neatly, you can make a neat frame and access panel, someone will have to get in there in the future. You can also disguise such an access with a large air return grate. People will see it and ignore it, thinking it is part of the A/C.
It's a nasty job... wear rubber gloves.