New to Computer Audio, Need Advice on a Headphone Setup

I'm pretty much completely new to all these modern computer audio configurations; I don't even know what a DAC is. Until recently, I was under the impression that sound cards had become snake oil and the last time I bought a sound card was probably over a decade ago. I currently use a cheap $15 Logitech headset and my onboard Realtek audio. I am building a new system soon, so this is my opportunity to improve. I keep hearing from people that using a sound card is better than onboard audio; and that it could help me in situations like watching an action scene and hearing distortion at certain points.

I should be very clear that the primary purpose of my desired audio setup will be watching video content, such as movies, with a small secondary purpose of listening to music; I do not plan to play games on my new system. I have no idea of what I really need to accomplish my setup, and my research has lead to a great deal of confusion so far. I do know that I want a good quality audio for watching movies and to a lesser extent listening to music, with as little distortion as possible.

I want a headphone setup, because I do not wish to disturb the people around me, and would like to have virtual surround sound, where applicable, especially in movies. For the headphone type, I prefer out of ear, closed, headphones, with a sub-preference toward full sized circumaural headphones but supra-aural is probably ok. I would like to accomplish this entire setup (headphones, sound card, etc) with a tight budget in mind. I could possibly dedicate as much as $200, but around $100 or even less would be ideal; perhaps you could make some suggestions in tiers, and explain what the beneficial differences are between the teirs.

Thank you greatly for any recommendations and information you provide me!

I think the STX would be more along the lines of my highest end option; I'm pretty sure for that to work in my case, i'd need to be able to buy a good set of headphones for under $60 to go with it. I'm pretty sure that'd be my highest end option, and probably not actually reachable for me right now. I'm not exactly sure how much I can throw at it for my new build, because it depends on how much I'm going to have left in my budget after the rest of my PC components in the build (which will be lean), and the total budget depends on my income tax return.

I was ideally looking for the whole audio setup (headphones, sound card, etc) to be around $100 or so. I'm ignorant to the ways of modern PC audio, so I may be asking for the impractical or impossible; if I am, let me know.