Well, first let's define 'classic' as a game that is at LEAST ten years old. Anything newer than that still has to earn its stripes.
IF YOU ARE A FAN OF STORY DRIVEN RPGS: Planescape Torment. Still one of the absolute finest stories in all of gaming. Engine is getting rough around the edges (it was never pretty) and there are voluminous amounts of text involved...this is a game you read as much as play...but it set a high watermark for fantasy narrative that still hasn't been equaled in the genre.
IF YOU ARE A FAN OF MASSIVELY MULTIPLAYER RPGS: There are three games that helped define this genre. The first and most obvious is World of Warcraft...if you like the genre, you've already played it, and/or a dozen clones of it. It was a title that blew past "genre defining" to become an industry defining hit and pop cultural crossover. The game is in its dotage now, but still astonishingly relevant in terms of market share. If you want to see the game that inspired WoW, look into Everquest...but not the game as it is now, look to play Everquest Classic, and re-create the late 90's experience. One of the most brutally punishing games ever made. And finally, it would be remiss to talk about the genre without mentioning Ultima Online, which might still to this day be the most ambitious MMO ever made (although most of its ambitions fell flat). It stands with some other aging titles as a reminder that when the genre was in its infancy, before the WoW clone template had calcified, that "MMO" could encompass a great many things beyond simply leveling up and tweaking your gear.
IF YOU ARE A FAN OF THE FPS: Either Deus Ex or Half Life 2. The former is a revered cult phenomenon praised for its open design and the "freedom" it afforded the protagonist, and is wrapped up in a ludicrous conspiracy fable. The latter is a well tuned master class in pacing and one of the earliest examples of telling a coherent story using environmental design and other indirect methods. I'll also mention System Shock 2, which is more of a FPS/RPG/Horror hybrid and thus difficult to assign a category to, but is a classic that shouldn't be missed.
IF YOU ARE A FAN OF STEALTH GAMES: Thief or Thief 2. Seriously though, if you are a fan of stealth games and you haven't already played Thief or Thief 2, what on earth is wrong with you? They are the Alpha and the Omega of stealth games. The rest are all cheap imitators.
IF YOU ARE A FAN OF TURN BASED TACTICS: The original Xcom is a truly special game, but the best pure tactics game ever designed has to be Jagged Alliance 2. In an age where tactics games have become clean and streamlined, often to promote tablet and mobile friendliness or make console versions palatable, JA2 remains a glorious mess; a pinata bursting with weapon and ammo types, a panoply of wildly differentiated mercenaries, and a huge map to conquer grid by grid. The story is pure tripe and an exercise in deliberately comical excess, but the tactical goodness is peerless.
IF YOU ARE A FAN OF GRAND STRATEGY: Total War: Medieval 2. Feels like a slight cheat as it's technically only nine years old, but it's close enough, and has been buried in time by a litany of warmed-over follow-up games that have diluted the franchise. If you're particularly fussed about its age, choose Rome instead...both marry grand strategy with RPG elements beautifully, and can devour hundreds if not thousands of hours. Tempting to nod to Civilization here as well, but we've been playing an ever-more refined version of that same game straight up to the present day.
IF YOU ARE A FAN OF SPACE COMBAT: Both the Wing Commander and X-Wing/Tie Fighter series were well loved...the former for its cinematic glory, the latter for its quality combat. Privateer was also a well loved game in its time, echoing back to the old Elite games (which were forgotten in the mists of time before we got a new Elite game).
IF YOU ARE A FAN OF SPACE STRATEGY/EXPLORATION: ...and have a strong stomach for retro, look into "Starflight 2", a hugely respected classic that is oft forgotten. For a slightly more modern take on some of its game play conventions, you can look at Star Control 2. Both tremendously enjoyable series that don't really have a present day incarnation, although Mass Effect was clearly inspired by both in small ways.
IF YOU ARE A FAN OF "SPORTS": Speedball 2 Brutal Deluxe is a cult classic...doesn't model a "sport" particularly well but had ridiculously fast action and sweet spot difficulty.
IF YOU ARE A FAN OF SIMPLE FUN/A STUDENT OF GAME DESIGN: M.U.L.E. is for you. An exceptionally old game that is still perfectly playable today, and one of the most influential games of all time...a litany of modern greats name check it as a game that got them interested in game design. Simple to play, easy to understand, and fun. I will also call out Sid Meier's Pirates! as a game that defies genre restriction and presents a glorious mess of different features and game types that all mesh together wonderfully.
IF YOU ARE INTERESTED IN RPG/GAME HISTORY: Check out any of the Ultima series (I recommend IV, V or VII) or the Wizardry series, they were titans of the genre in their day, and the former series pioneered a multitude of genre conventions that underpin virtually every modern series. You need a very strong stomach for retro gaming though, the earlier games in both series are positively ancient. If you think games like the original Fallout or Baldur's Gate are too old to get into, beware all ye who enter here.
Honestly I could sit here all day listing out classic games people should try, but ain't nobody got time for that.
