I'm glad to help! Yeah, I think you're looking in the right places. And yes, most of the headphones listed above are good "references" that will work great as starting points, and while eventually you'll look for something different/better you won't regret buying them (if you know what you're buying). Don't be one of these fools that buys a HD6XX and sells it straight after because "they have no bass, dull sound, boring, low volume from smartphone and laptop, my $20 Sony/Philips headphones sound better".
I won't recommend first getting an amp, unless you already have (or will have really soon) at least one headphones that really benefit from proper amplification. Otherwise, you're just wasting money, because it won't make a big difference in sound and you'll do better buying better headphones with the money you spent on the amp. So, first of all, buy some good headphones. Ideally, you'll figure out your desired headphones based on your preferences and the kind of music you listen the most, forgetting about if they need amplification; once you know what will your first headphones be, if they're not easy to drive, buy a proper amp for them.
If you're 100% sure you wanna get into the audiophile world, I'd suggest you to save at least $250 (plus your headphones budget), so you can buy dedicated source (like a DAC, or a CD player, or a turntable, or...) and amp. Why? Because later on you'll probably want to upgrade your setup, and having discrete DAC and amp allows you to upgrade one without having to replace the another. A less risky approach is buy a cheap DAC/amp combo (like the Schiit Fulla 2, Audioquest Dragonfly or Hifime Sabre 9018) and use it later as a DAC; this won't be as good as a dedicated DAC, but It'll be better than the typical headphone/line out that you can find on most devices like PCs or smartphones.
To sum thing up, let's finish with a case study. Based on the opinions and reviews, you decided that the HD6XX are good for you. Since the HD6XX have high impedance, and knowing that your actual sources (let's say an average laptop and smartphone) won't deliver enough power under such load, you have to buy an amp: you can go easy and buy a DAC/amp combo, which will be close to the "full package" but cheaper, or go big and buy separate DAC and amp (usually called "stack" because they're usually meant to be placed one on top of the other, like a Schiit Modi 2 DAC & Magni 3 amp on top). The decision is yours, but if you can afford it, the stack will eventually prove worthwile when you decide to upgrade the DAC or amp.
I started with a DAC/amp combo, the Hifime Sabre 9018, later bought a Little Dot I+ hybrid amp with some upgrade tubes, and finally a Topping D30 DAC. This is my actual stack, and works great with almost anything: from low-impedance low-sensitivity planars like my T-X0 to high-impedance high-sensitivity HD6XX. For IEMs I prefer the Sabre, because it has absolutely zero background noise or hiss and has more than enough power to drive most of them, but for the rest the stack is way better. Actually it's been like three months since last time I used the Sabre, because when I listen IEMs on-the-go I plug them straight to my phone and it sounds ok and has just enought power with most of them (S7). But I'm not you, so YMMV.
Wow, long post again hahaha. I should start a blog or something! Well, good luck and feel free to ask anything else. I hope my experience is useful for you, and helps you build a good starting setup :)