I’d keep it simple: plain HTML + CSS is enough for this style, and you can learn a lot just by recreating old layouts by hand. Use a text editor, fixed-width containers, tables or table-like layouts, small GIFs, beveled buttons, tiled backgrounds, and system fonts. You really don’t need React or anything modern for that look.
Since you mentioned hosting animations and downloadable games, I’d also think early about file organization, download pages, and navigation. A lot of old sites felt “right” because everything was easy to browse even when the design was busy. This kind of structure matters on bigger content-heavy sites too, whether it’s a personal retro page or more complex projects like travel website development services, where people need to move around a lot of pages without getting lost.
So yeah: start with HTML/CSS, study old layouts, and build one simple page first before adding extra sections.