For your purposes, any hard surface in the room can be considered highly reflective across the spectrum of sound. As you've noted, with lots of hard surfaces the sounds reflect and persist long after the source of the sound has stopped. This is known as decay, and plays a very large role in shaping your perception of sound in an enclosed space. The goal should be, in my opinion, to allow enough of that reflected sound to persist for you to hear it, but not so much that it overpowers other sounds. At the same time, you want to avoid situations that lead to flutter echo and other position-dependent anomalies.
In contrast to reflective surfaces, most any fibrous material will absorb sound, to some degree. Rock wool, cotton batting, insulation, and even upholstered furniture can all absorb sound and diminish decay times. Here's where it gets a little technical and unintuitive: sound absorption effectiveness is sensitive to wavelength. Low frequency sound has a very long wavelength and can therefore require very large piles of fibrous material for absorption. That matters because if you add thin layers of rock wool to your walls, you only absorb high frequency sound. Low frequencies are allowed to persist and they overwhelm your ability to hear subtler sounds, so as you add absorption to your space, you want to do it in a way that maintains the spectral balance of your music. In regards to the location specific anomalies, mostly that's flutter echo caused by parallel reflective surfaces. I'm sure you can hear a "zing" or spring type of sound if you snap your fingers, clap, or tap in the room now. You can clear that up by making sure that the absorptive material you add is distributed across at least one of any two parallel surfaces - even better if it's on both, but not covering them completely. Since the floor will be hard, make sure you get some on the ceiling.
I suspect that adding absorptive panels like I described - about 4 inches thick - to walls and ceiling will be adequate. You probably won't need any diffusion, but it can be useful. For instance, if you use more and more absorption to control low frequency decay, you can end up absorbing too much high frequency. To correct that, you can add diffusion that will redistribute high frequency sound around the room to restore the spectral balance.
It's nearly impossible to do this without measurement equipment. You can probably use mics and cabling you already have, but you will need software like the program I mentioned earlier.
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