Hi TSasquatch!
If you aren't willing to purchase an amplifier and want to run off of your smartphone then the K371s isn't a bad place to begin your audiophile journey. They were actually my first; though, I did end up returning them. For the price point, they have really good detail with a little warmth and musicality. Personally, I thought they out-performed the Beyerdynamics and Audio-Technica with similar price points, but I'm also just not a fan of those brands' V-shaped signatures. Plus, they're reasonably comfortable, portable, and (while largely plastic) still have a fairly premium feel to them.
All of that to say, I did end up returning them. I did not find them aggressive in their presentation to music, but they also have a reasonably small sound stage with a fair amount of detail packed into that. So, as the music I listened to go more complex, the more rapidly ear fatigue set in. Listening to small band, acoustic-based stuff or jazz was great, and I could listen for a long time. Yet, listening to albums, genres, etc. where there is just a lot going on (e.g. symphonies, scores, some rock, etc.), I'd tap out from the headphones after like 30 minutes. Now, keep in mind two things - I have fairly sensitive ears and what I've just described goes over a lot of people's heads. My good friend, who would not consider himself an audiophile, basically just goes "I have no idea what you're talking about, these just sound like great headphones to me".
If you watch Zeos' review of the K371s at ZReviews on Youtube, he describes well a similar experience to my own. He would actually recommend the K361s over the K371s, which I think is going a little far. Other reviews would tell you, however, that they honestly believe every "audiophile" should own a pair of K371s given their price to performance ratio.
I've also tried the K553s. Build quality is great and are very durable. Sound quality is good, but nothing to brag about. Comfort is okay; again, good enough but nothing to brag on. I found the K553s are sort of like an "ole faithful" type of headphone; well-rounded and dependable, but nobody is going to go "these are the best headphones ever" because they just don't have any super definable features to them. They do, however, out perform the K371s in their build quality (considerably more metal and sturdy construction) and have a more open sound than the K371s. Like you, I've got a fairly eclectic music taste and I found the K553s to be a better all-rounded than the K371s. I only really enjoyed listening to some jazz and acoustic-based indie/folk stuff on my K371s because of its sound signature. I thought the K553s we're better balanced and so could still play jazz and acoustic stuff well enough to enjoy it, but could also do genres like rock and classical far better.
My favorite closed-back to date are the Meze 99 Noirs, but they aren't for everybody. I like the build quality. Comfort is fantastic. They have a smooth, warm, laid-back sound signature - the perfect late-night, kick back and relax and listen to some smooth jazz type of headphone. But, they can be a little bass heavy. The bass is tight and doesn't bleed into the mids, but its definitely pronounced. I have found that if you use a 2.5mm balanced cable that it does tighten up even more, but that also requires you to purchase an amp. Those also retail for $200 (USD) on Drop.com.
Basically, the K371s are a good headphone, but I would also look into the K553s as a strong competitor and look into the 99 Noirs, if they sound like something you'd be interested in. I would note, however, that I have found it very difficult to find a closed-back that does eclectic music listening well. I had a similar budget to yours and I ended up going with the Meze 99 Noirs because their smooth, bassy warmth hit really well for my acoustic, jazz, and rap tastes which is about 75% of what I listen to. Then, I got a cheaper pair of IEMs (KZ ZS10 Pros or ZSX "Terminator" are good options) because they had a more lively presentation for when I wanted to listen to rock and stuff. Whatever you decide on, I also highly recommend looking into getting the BTR5. It's a bluetooth/USB DAC/amp combo that gives many DAPs a run for their money. None of the above mentioned headphones need it, but the BTR5 will produce better sound than what comes out of most smartphone headphone jacks.
