If you are into headphones, you could start hanging out on the subreddit. It's a big community, and you'll see many of the current technologies. It could get really expensive, I mean the new Sennheiser closed-back HD820 has a ~$2k price tag (ouch!), but certainly you could call yourself done with a $400 dollar setup, including DAC and Amplifier.
EDIT: You could also try IEMs, Etymotic makes some of the most clear, coherent, natural and neutral sounding. Their TOTL (Top-Of-The-Line) is the ER4 line which has a MSRP of $350, but you could get the same value with the ER3 line at $179.
The first thing to do if you want the best quality sound is to sort out the acoustics in the room you're in. Do that before buying or building any high end gear, otherwise you're just throwing money away. Shit speakers in a well treated room will sound better than the most expensive ones in a poorly treated one.
Get yourself a reference/measurement microphone so you can measure the room, then use REM [1] to analyse the results (it's free). The Behringer ECM8000 is fine ($50), you don't need to spend a lot of money on that. You put the reference mic at your ideal listening position (you're not going to make an entire room acoustically perfect without needing a second mortgage, even high end studios aren't).
What you want to get to is a flat response across the frequency range. You won't get it, and you will need bass traps, mid traps, etc. to fix the problems. If you're doing it for your TV then you can get attractive looking traps that don't look ugly. But if you're doing it for music, and you want lots of bass, then you may end up with big bass traps in the corner of the room. Although the PSI Audio AVAAs [2] are another (more expensive) option for trapping bass without taking up any more space than a regular speaker. I trialled a couple of these and they were exceptional.
Depending on how bad the room is you can use room correction systems to fix it. I use a Trinnov ST2 Pro [3] (on top of a treated room) in my studio and the difference is astonishing.
This is my frequency response before and after the Trinnov [4]
Once you have got your room acoustics to a pretty good place, then the world's your oyster in terms of audio gear and speakers. I would advise against getting trapped in the super expensive audiophile world, you won't get much difference once you're above the $10k speakers range, and actually you can get results which will blow your mind with $1k speakers. Building your own is tough, because the dimensions are so critical, but if you're up for the challenge, it could be worth it!
I've spent a large amount of time in this world because I've built a high-end music studio in my house. And going much beyond this advice isn't worth it. You need to go and listen for yourself and work out what you like. But acoustics will always win over gear, always.
Any specific beginner-friendly resources that you would recommend?
I thought of starting with something small and defined, like upgrading my TV's sound but even that comes with a variety of bewildering options.