Welcome cannon_dt,
Speakers are the only transducer in your system (something that changes energy from one form into another), have to work in unknown environments, and are the only component "really" involved with physical movement so have the toughest job of any component in the audio reproduction chain. Some speakers, like Dynaudio can be very inefficient, so sizing the amp is important and can only be done once the speaker selection has been made. Some speakers, again like the Dyns have a low impedance rating and so need a more stable amp. And of course some speakers provide deeper bass than others, which affects the type of amp that would work best. Since they do all those functions quantifying their quality is hard to do and so speaker selection ends up being a very personal matter. Therefore always shop speakers first.
From your descripton is seems that you're suffering from listener fatigue. This is the result from the brain working too hard/long to "make sense" of distorted sounds. The list of possible distortions and reasons for them is long. Some speakers are more "listenable" than others meaning that they are not very fatiguing. No doubt the number 1 source for listener fatigue is playing the system too loud, either beyond the rated speaker/amp capacities, or your ear/mind to absorb further stimulation. For me speakers with cheap drivers, metal cone/dome drivers, higher order crossovers, or exagerated frequency response (too much emphasis on a given frequency range) are the worst offenders. Speakers that are less detailed (like simple built-in TV speakers) are often considered more listenable. When you think about it, TV speakers don't exagerate frequency response, aren't metal, or have any crossover issues.
If you're limited by room size or how far the speakers can be moved out into the room, you should avoid speakers that have really extended bass response and horn loaded midrange drivers or tweeters (that can seem to be very much "in your face"). Your need to know your preferences in musical genres, your room size/equipment layout within the room, what audio attributes interest you, and of course budget before effectively shopping for speakers (or we can help much with specific recommendations).
Be warned that the first/biggest mistake audio shoppers make is being initially "wowed" by what they're hearing. That "wow" factor is typically boomy bass and/or (zingy) treble frequency response and, as mentioned above, is a common reason for listener fatigue. Take your time auditioning and of course do it without distraction. Use music that you're familiar with and that represents your interests. Limit yourself to auditioning 3, maybe 4 speakers per day as the mind can't "seriously" juggle more than that. Try hard to listen at home too before making the final purchase.
Take care.
