Disclaimer - I’ve by and large only played Horde, so take that as you will
Generally, speaking, Humans and Dwarves have the most well put together story in Classic. Most of Classic’s endgame focuses on events involving Blackrock Mountain and the associated factions, and these tend to be fairly Alliance-centric. The Horde does have its own parallel stories, but they don’t really compare to things like the Alliance Onyxia questline.
Now, there is also an important caveat for Classic stories as well - a fair chunk of the late game stuff is also done by neutral factions. These tend to lean Alliance (Thorium Brotherhood, Cenaurian Circle, Argent Dawn), but the overall questing experience with them is the same for each faction. So keep that in mind too.
(I can’t speak from experience from this, but humans go to Westfall, Redridge, and Duskwood, while Dwarves go to Loch Modan and then Wetlands(?). So, I think they have a bit more of a unique experience than the Horde races.)
Now, as far as races go, not all of them get the same mileage. Gnomes and trolls are stuck piggybacking the Dwarf and Orc starting areas respectively, and never really get much content of their own. Beyond this, the three Kalimdor Horde races are pretty well funneled into the Barrens and then all of its spinoff zones going around setting up outposts and handling local threats. There really isn’t a grand overarching story to any of this though. The Forsaken have a fairly consistent narrative, but it ends with Hillsbrad. Oddly, they don’t continue on in the Plaguelands (I always considered this a mistake on Blizzard’s end). As for the Nelfs, there’s a lot of Nelf ruins and history in different parts of Kalimdor, but it never really comes together in some sort of grand plot - you’re still going around and solving local problems for your quest hubs and outposts.
Humans and Dwarves are somewhat the exception to the above, because the Defias story and Dark Iron Dwarf stories both continue up until the endgame dungeons (though there may be a bit of an intermission where you go do other stuff elsewhere).
Ultimately though I’d approach things from the perspective of what faction and aesthetics do you like the most. Classic takes a while to play through from 1-60, and I don’t recommend playing through it on a race or faction you don’t like, all for a few quests at the endgame.
(For a bit of Vanilla Backstory, they did Eastern Kingdoms before Kalimdor, and the Gnomes and Darkspear were the last races added to the factions. So that’s why those two races just share other racial starting zones, and part of why Kalimdor doesn’t have the same level of depth for stories that EK does.)