There is a story and I was quite invested in it. However, it comes with caveats. It is difficult to experience the entire Warcraft storyline in a timely manner.
First, the story doesn't begin in WoW, but in Warcraft 3. Warcraft 3 introduces the factions, characters, and subplots that WoW revolves around. It introduces Thrall, Grom, the founding of Thrall's Horde and Orgrimmar, introduces the Night Elf empire and Arthas and the Scourge and Illidan and the Burning Legion and so on. The WoW game doesn't really do a good job introducing you to those characters. It does an excellent job introducing you to the world, but when you finally start doing expansion storylines that revolve around Arthas, Illidan, Jaina, Thrall, etc, WoW presumes that you already know who those people are and what happened in WC3, so it can be confusing.
The other issue is that a lot of WoW's story doesn't happen in the actual game. A lot of big, major events happen in books, many of which are out of print, or short stories that are hard to find on the official website since it's been revamped so many times. For example, Thrall becoming unpopular with the Orc populace and being pressured into appointing the young, popular Garrosh Hellscream as the next leader happens in a short story and a book. Cairne challenging Garrosh to a duel over the leadership of the Horde and then dying happened in a book. Garrosh nuking Theramore and causing Jaina to nearly wipe out Orgrimmar with a tidal wave happens in a book. Garrosh being put on trial happens in a book. Illidan being retconned from an evil tyrant into a good guy all along happened in a book. Sylvanas' motivations for starting a war happens in a short story and in a book. And so on. Right now, in the current Dragonflight storyline, the motivations of the main antagonists, the Primalists, are not well conveyed ingame, and you really need to read the new book to understand why they have such a seething hatred of the Dragon Aspects and the Titans.
I enjoyed a lot of the early Warcraft novels. The War of the Ancients trilogy, Lord of the Clans, the Beyond the Dark Portal novelization, etc. The authors back then had a lot of passion for Warcraft and were given a lot of leeway to make the most enjoyable book possible. The books really helped humanize certain characters too. But later novels are hamstrung by corporate mandated requirements. Christie Golden keeps trying to give Anduin a wife and Jaina a husband in the novels, but these aren't reflected ingame. By the time of War Crimes and Traveller, I wasn't really enjoying the new books anymore. A few years ago I decided to clear out my bookshelf of books that I wouldn't reread, and threw out all of my Warcraft novels. I kept only Tolkien, Narnia, and a few others.
Also, a lot of important storylines have actually been removed from the game over the years. The Battle for the Undercity which galvanizes King Varian's hatred for the Forsaken and the Horde was removed. The MoP and WoD legendary questlines which told the second half of those expansion's storylines have been removed. The Golden Lotus questline was removed when they were wiped out by Garrosh' nuke in 5.4. The BFA prepatch War of Thorns questline in which the Horde invades Ashenvale and Darkshore and then burns down Teldrassil has been removed. And so on. It's a shame because these were among the best questlines in the game, but they are no longer accessible. Reading about them on Warcraft wiki or watching a summary by Nobbel or the Lorerunner on Youtube isn't the same enjoyable experience as those storylines were.
Another issue is that a lot of important later storylines are locked behind lenghty reputation grinds. For example, seeing Malfurion and Tyrande being separated requires you to grind rep with the Maruuk Centaur, which can take a few days or weeks of doing dailies.
Lastly, you will need multiple characters to experience the story. You need at least one character on Horde and on Alliance. For example, in order to see the invasion of Gilneas storyline, you first create a Worgen character and see the fall of Gilneas from their perspective. Then, you need a Horde character and do the Silverpine forest questline, and see the invasion of Gilneas from their perspective. In Legion, each class has its own questline and some of them are pretty important like the Priest's questline, in which they acquire a sentient dagger that is going to be the big bad of the next expansion. Lastly, there are some race exclusive heritage quests, but only the Orc one is kinda important.
I think you kinda had to "be there" for Warcraft's story. Maybe you can get into it this late, but the story is now very, very long, and that would be a humongous time requirement, I think. I think if you want a good story experience in a video game, you would be better off playing shorter JRPGs like Final Fantasy or visual novels like Muv-Luv or Aselia. But WoW also has or had a lot of other nice things going for it in addition to the story, such as its aesthetics, character customization, the music, etc.
Another thing to watch out for is that until very recently, most of WoW's ingame storytelling was presented in paragraphs of unvoiced quest text, which most people didn't read. Recently they've been having more ingame cutscenes, which does help make the story more digestible.
If you want a list of the best storylines they are:
- Death Knight starting questline in Eastern Plaguelands
- Crusader Bridenbrad questline in Icecrown, a popular fan favorite.
- The invasion of Gilneas/Worgen starting zone (followed up by Horde Silverpine Forest questline)
- Stonetalon Mountains questline as Horde. You enlist in an army and rapidly rise through it's ranks as you fight Elves in the name of the Horde.
- The Wandering Isle/Pandaren starting zone
- The MoP levelling questline was overall enjoyable, particularly the Jade Forest, Valley of the Four Winds, and Townlong Steppes.
- The 5.1 Dominance Offensive/Operation Shieldwall questline was great, but it is timegated by a rep grind.
- The Isle of Thunder questline
- The Warlords of Draenor questline was great. Likeable characters, threatening villains, lots of action, visually fantastic environments and the best OST the game ever had. Definitely the best expansion questline IMO.
- The Legion class order hall questlines really varied in quality. The Death Knight one had some exciting moments but also really mindboggling writing.
- If the War of Thorns BFA prepatch questline ever comes back, absolutely do that on both sides. The final quest on Alliance was one of the most memorable experiences I ever had in a video game.
There might be some others I'm missing. There were other questlines that I found enjoyable enough, like Azshara or Highmountains, but I wouldn't put them up there in the must-play stuff.