i’m going to but in here and than you for this advice. I’m having a house built. Several weeks ago when they completed the drywall in the room which I plan on using for musical pursuits, TV and such, I clapped my hands and boy did it ring. I’m glad you mentioned about furniture and carpeting can help tame it. I still think I’ll need some traps as well as partitions. I’ve never heard of a gobo. Guess I’ll have to google it.
You'll want to look at acoustic diffusion, not absorption. Those foam eggcrate panels are next to useless, and in many (most) cases, do more harm than good (only absorb sound frequencies in a narrow band, which actually emphasizes the frequencies you're trying to control).
If you really want to use sound absorption, go with Rockwool panels, at least three inches thick (seven inches thick would be better), in combination with diffusion.
.