For reference, I come from the Brood War scene, and before that from the Command & Conquer games. I briefly played SC2 Wings of Liberty and didn't particularly care for it, and I never really cared too much for the MOBA (LoL/DotA) games. That should give you some idea of what I probably would or wouldn't like in this game and where my commentary is coming from. I haven't ever played the WC3 expansion (The Frozen Throne) but I'll definitely get to it eventually. I'm sure a lot of you are far more familiar with the game than I am, but nevertheless I want to give my impressions as I played through it.
With a whopping 34 missions, this campaign is definitely a hefty one. It follows the story of all four playable races - humans, the undead, the orcs, and then the night elves - with a consistent narrative told throughout it all. Since I'm generally interested in the storytelling of it all, I took a look at how the lore is laid out relative to the game's predecessors, and as far as I can tell it's essentially a standalone narrative in the same continuity as WC1/WC2. That's good, I'm not really up for having to play the predecessors just to be able to follow the story of this one.
Obviously the only way to play is to complete the campaign on the hardest difficulty setting, so that's how I played. Though my strategies tended to be extremely simplistic due to the fact that I simply wasn't interested in having to learn complex micromanagement maneuvers and spellcasting systems as are present in professional play. My army compositions usually consisted of basic infantry - riflemen and footmen, grunts and headhunters, so on and so forth - along with the hero units and artillery (catapults, mortars and steam engines, etc). It usually worked alright, with some notable exceptions, though I must say that in subsequent parts of the campaign I often had my ass handed to me by units I never learned how to use. After I finished Human and started playing Undead, in the later missions I continually got brutalized by sorceresses, which I didn't use because of the painfully annoying mechanics they had. Though nothing hurt nearly as much as the sheer beefiness of the Orc armies...
If you need a reminder of what the missions are, here's a list that should give you a decent synopsis of them all.
Not much to say because it's really cookie cutter, but it's definitely a nice touch that it has a story element to it. Gave me a reason to play it, at the very least.
Following the story of the clearly not villanous Prince Arthas, this game takes you across the great human kingdom of Lordaeron as the prince and his troops attempt to save the kingdom from the undead plague. By the end the prince abandons everything and joins the Undead as his pursuit of victory drives him insane.
Honestly, this part of the campaign is also very simple and I don't exactly have too much to say about it. The first few missions are tutorial-esque, teaching you to build units with the human building mechanics, and to scour the map for NPC opponents ("creeps" I believe they're called in WC3 lingo). At any rate, all of the standard missions were pretty simple mass-and-destroy affairs, where a standard big army composition was more than enough to break through the enemy's defenses and take the win. The fifth mission and the final mission also had a significant defensive component to it, a task which is relatively easy to perform because the human structures are quite beefy. The units were mostly nothing special (just your generic melee/ranged/spellcaster units), with the exception of the steam engine, a short-range heavily armored building killer. Seems like humans would be a great race in a game where you aren't highly incentivized to have smaller unit counts by the low maximum supply and the upkeep mechanic.
It's kind of unfortunate that the best hero you have (Jaina/Archmage) only joins your team for a short time. The result is that you're left mostly with heroes that are usable, but that are really fairly minor presences on the battlefield. Ultimately the Arthas/Paladin hero is very helpful with the Holy Light (heal/enemy damage) spell, but healing standard rather than enemy units is not really all that great. Meh.
The only mission that gave me trouble here was the final one, where Arthas gains Frostmourne and challenges the big villain of the human campaign: the Dreadlord Mal'Ganis. And really, the only difficulty in this mission is dealing with the fact that it's the first time that the enemy sends their hero into battle.
Arthas revives as an undead, resurrects a minor antagonist from the previous campaign segment (Kel'Thuzad, who turns from a nobody into an actually decent character), revives a powerful daemon, and destroys lots of people in the process with the weaksauce power of the Undead army.
The Undead units are mostly really brittle and there's really not much I can say about it. The Necromancer represents one of my favorite designs for a spellcaster and it's a real shame that it is nigh-unusable simply because of how dependent it is on high mana and the unrealistic presence of well-positioned mounds of corpses. Ultimately I ended up solving most of the problems with Crypt Fiends, one of the most impressively versatile units in the game that has a great counter to air-based foes and a not-so-bad mix of HP and offensive stats (and is also a really cool unit design). Some problems could only be solved via Frost Wyrm, but most everything else I did using mostly armies of massed Crypt Fiends. I actually really liked the Undead and their unit composition, but it just doesn't seem like they hold up well against any of the other races - which seems to be the case competitively.
The heroes are actually pretty cool though, with Death Knight Arthas being a slightly more usable clone of his Paladin self (mostly due to the added mobility with riding horseback), and Kel'Thuzad/Lich being a useful spellcaster. Better than the human bunch, to be sure.
I felt the design of each of the missions was fairly simplified because as poorly as I played, I got through this set of missions in relatively short order. A group of twenty Crypt Fiends with heroes and artillery mostly killed off any foe that didn't just throw themselves upon a wall of Spirit Towers and died that way. Only the last two missions posed any threats. The seventh (second to last) was tough because the auras made any approach difficult, and the blue base (behind the second aura) was so tanky that I really had no means to break it without abusing the AI to lure out the hero and Frost Wyrms to freeze the base's production (all while having to defend from two bases worth of beefy air and ground attacks). And the final mission because it involved relentless attacks from multiple bases that could only be deflected by Frost Wyrm attacks. Many a time I lost on this last mission with a second or two to go, until I realized that there was nothing stopping me from just casting Death Coil on Kel'Thuzad to extend his lifespan by about that long .
The green dudes that are addicted to combat have to travel west to protect themselves from the daemons, and have to suppress their desire to kill stuff long enough to work with the humans to protect their race from the approach of their daemonic foes!
The Orc campaign is actually a sort of interesting one, with quite a few no-base missions with limited units that relies on efficient hero use. Really, the hardest part of this campaign was to fight air unit swarms (especially harpies) because the orcs' most efficient way of doing battle against air units is simply to throw down guard towers. Overall, Orcs felt to me like an upgraded version of humans, with weaker defensive abilities (weaker structures, no spamming peons to build a building quickly) and no steam engine, but with vastly superior offensive capabilities. Not a single mission was actually particularly difficult because Orc units are just so powerful. Interestingly, the only difficulty in the final mission was the Orc opponent units themselves. The undead and daemonic units? Easy pickings.
All the heroes had pretty good special abilities, but of course by far the best is Grom/Blademaster, with a hefty damage output and the brutal Bladestorm attack that takes out gigantic swathes of enemy units. Basically there was no ground threat that ever made life difficult for the Orcs, by virtue of beefy melee units, a powerful siege artillery, and OP heroes. Only air units were ever a problem, and the campaign never really did have too many of those. Easy campaign, though this part of the plot is the best so far in that it covers a wide range of the characters that play a role throughout the whole series.
Orcs are beefy and strong. What else is there to say? OP race.
Night Elf Campaign
The blue elves who love nature and hate anyone and everyone else fight in the name of a gigantic horse man against everyone who steps foot inside their forest!
I basically played this campaign with nothing but archers and the occasional ballista. I briefly tried diversifying my strategy before coming to the conclusion that Night Elves have by far the best massable basic infantry and that the computer opponent didn't really warrant any other approach. Two missions proved to be sort of difficult for me: the third and the final mission. The third mission, awakening Furion, was tough because of the time limit, it took me a few tries to be able to get past the Orc base quickly enough to meet the time required (had to learn to spam ballista shots at key buildings more effectively and to create more archers). The final mission was a doozy, though: defending the World Tree against constant assault by an all-out daemonic/undead foe, with barely functioning human and orc defenses, for 45 minutes. I have to admit that after spending four days on this mission and trying out quite a few different strategies, I used a cheat code to get the ending - before instantly regretting it and trying again with a strategy that finally worked: archer/dryad spam, save between waves, run away from the base as soon as it becomes clear that there is no way to survive that wave, and generally just stalling for as long as possible against this kind of threat.
All the heroes here are pretty good; the two main heroes have the Starfall/Tranquility Ultimate combination, which makes for a tanky and deadly defense in the final mission. But Illidan/Demon Hunter is by far the most awesomest of them all. Undeniably useful given that this is the most important hero in competitive play, but also just has a great character design. Only present for one mission, but in that one mission he takes out the second most powerful opponent of the entire campaign (Tichondrius) as if it were just no big deal. Too bad that everyone just instantly hates him.
At first I didn't really like Night Elves - their combat mechanics were bizarre and they seemed a tad irrationally xenophobic - but after a while I started to appreciate them. They have some interesting mechanics and spellcasters that, with practice, could definitely make them a really cool race to use. Being the final campaign of the game, they got the hardest mission of it all - but it's a pretty neat race.
Other Thoughts and Conclusion
I gave quite a few thoughts on individual aspects of the gameplay throughout the Campaign section. I didn't devote all that much time to multiplayer, since Frozen Throne is the one that's actually played competitively, but a few things did strike me. I did watch a few pro games though to get a feel for how 1v1s are actually played, most notably this one:
Honestly, with all the mechanics of this game, I have a hard time calling it an RTS in the purest form. Between upkeep, a supply cap that makes it harder to have more than about thirty combat units (and more realistically 10-20), hero units, and most importantly the prevalence of RPG-esque elements like random foe encounters, leveling up heroes, and frequent item use, it's more like a hybrid between an RTS and other genres. I suppose it's a great testament to that fact that that is exactly how the successors of the game developed: the hugely popular MMORPG World of Warcraft built further on the RPG-like elements of the game and the storyline into an RPG proper, and the MOBA games (DotA and its successors) retained something of that hybrid style but focused even further in on that hero-centric mechanic.
None of that makes it a bad game in the slightest; on its own merits this actually turns into a pretty neat game. I sort of see a lot of WC3-esque choices made in SC2, a definite wrong choice since Starcraft is more of an RTS proper than the hero-centric and sandbox-like WC3 mechanic. But in its own way, this game makes for a very interesting spectacle to watch and a not too bad game to play.
Still, it's sort of interesting. Despite still having a definitive fanbase, however reduced over the reality of aging with the times, this game is very much notable for the way it inspired some rather unique successors with its unique twist on the RTS mechanics and its support for custom content. Ironically the game it helped create (WoW) and the genre that was built upon it (MOBA) have eclipsed WC3 itself in popularity. Yet as a board that largely started and functions as an RTS community, the original certainly seems to have a fandom all its own.