Full disclosure: I have never watched Ten Things I Hate About You, nor could I be induced to do so absent monetary gain or coercive measures involving hot pincers and a rack.
Helpful NPC Thom here to talk about the Fate RPG. The full game is available online, so take a gander at your leisure. I'm going to discuss the system at length and share my perspective on where the game's strengths and weaknesses lie from the perspective of a Very Traditional GM. While Fate is billed more as a story- or narrative-oriented game, its basic structure is traditional, and that's one of the reasons that I have taken to it over other storygames.
Before You Begin:
Read the Book of Hanz. It's like those 400-page Prima guidebooks for shitty console JRPGs, except it's for Fate. Without the Book of Hanz, Fate is going to place the Befuddled Aspect on you, due in part to Fate upturning game expectations from a traditional roleplayer's perspective and due in part to Fate (like most story-slash-narrative-oriented games) does a resoundingly piss-poor job of explaining itself.
Fate Is Not a Roleplaying Game
Recognize that Fate is not a roleplaying game insomuch as it is a roleplaying system. (I'd argue that it's a storytelling system, but let's not quibble semantics.) A good example of a roleplaying game is the masterpiece that is Tom Moldvay's Basic D&D: its rules dictate a clear structure of play with defined procedures (and a win/loss state).
Contrasting this, Fate's mechanics are all tools for storytelling. It's up to the GM (and the group) to know how (and when) to use those tools. D&D gives the GM a hammer. Fate gives the GM a toolshed stocked with a full complement of Black and Decker tools. Fate gives a plethora of ways to use mechanics to best represent the story your group is telling.
Fate sort of offers guidance on character creation and the results of rolling dice, but it doesn't explain how to play.
As an example, take D&D and fightin'. Players and GM know exactly how a fighter engaging an orc in combat works: cue Final Fantasy music, roll initiative, roll 1d20 + bonuses vs. Armor Class, roll weapon damage on a hit. Simple and straightforward. Although Fate has its own version of the attack roll, the concept of "an attack" is more nebulous. Characters can make "attacks" against anything because anything in Fate can be structured as a character with defensive abilities an a health pool (Stress track). Mocking someone can be an attack; a doctor injecting antibodies might be an attack against an infection; a lawyer arguing his case might use his evidence as an attack against the defendant.
Amorphous mechanics are Fate's wheelhouse. This can be confusing. All of Fate's dice rolls come down to one of four Actions: Attack, Defend, Overcome, and Create An Advantage. In theory, this is simple. At the table, it's another matter. Fate's mechanical fluidity contrasts with the rigidity of other roleplaying games. A dice roll in Fate has a corresponding Action varying based on the character's intent and the level of narrative detail desired. In Fate Core, this is not well-communicated. (Fate Condensed may express this more clearly, but I haven't read it.)
Narrative Lens As Part of the Toolkit
The general consensus within the Fate community is that the subsystems are pacing mechanisms. Disagree. Not entirely, mind you, as there's a definite hint of truth in there. However, as my writing has alluded, I view the subsystems as a form of narrative lens, allowing the table to zoom in on important scenes while zooming out on less important matters. This alters the pace of gameplay, but I view the primary intent of those mechanics as creating different levels of granularity; thus, narrative lens is the cause and the effect is pacing, as opposed to the other way around. If the authors disagree, I Compel Death of the Author so they cede to my argument. (Death of the Author is bullshit, mind you, and I'm going to invoke Sick of Literature Classes to bolster that statement.)
Where was I? Ah, yes, granularity. Fate's mechanics range from a wide scope overview all the way to a detailed, moment-by-moment narration. It might be helpful to think of them in terms of sentences and paragraphs: the more ink spilled to describe something, the more it moves from a single Overcome to a Challenge to a Contest to a Conflict.
Example: The Lord of the Rings
Allow me a moment to demonstrate this in practice. Consider a game inspired by Lord of the Rings. In this style of game, the journey is the adventure. There are moments where fighting takes place, but the majority of the story is travel, song, hiding, and remaining hopeful against despair. In a traditional RPG, the hobbits departing the Shire and evading the Ringwraiths would likely be a single test, the mechanics defaulting to a single Stealth roll (or another analogous skill). Fate offers more in its storytelling toolbox. Depending on how invested the table is in that as a scene, the GM could structure it in a few ways.
A single Overcome roll. If the table isn't that invested, the entire scene could be narrated with one roll.
A Challenge. The GM opts to structure the scene over a series of rolls. First, the hobbits have to test their Athletics skill because it's a long road to Bree. Then they have to test Stealth to evade the Ringwraiths. Then they have to test their Rapport to convince the Bree watchman to open the gates for them. The flight from the Ringwraiths thus becomes a smaller part of the overall narrative.
A Contest. If the cat-and-mouse game is more central to the session, the GM could call for a Contest, where the hobbits are rolling Stealth against the Ringwraith's Notice multiple times in a race to score three victories. (This is probably the best way to structure it.)
A Conflict. If the table really wants to dig into the chase, Fate allows the chase to be structured akin to a traditional RPG's combat mechanics. I could see this working with the hobbits having a group Stress track (hit points) called Discovery and the Nazgul having their own called The Trail. The two exchange Attacks, depleting the respective "health bars," imposing Consequences like Thrown Off the Scent, Trapped, Yearning for Home, Displeasure of the Lidless Eye as they strive to overcome on another.
Each leg of the journey could utilize different mechanics to demonstrate the importance of narrative weight.
Fate Demands Customization
Fate is a complete system out of the box. You can play Fate with the barebones rules. Personally, I don't find it satisfying without any crunch whatsoever. Fate's premise is that you can tweak the "dials" of the game to customize it for your group. Without weapons, armor, and magic rules, the game is missing something. Your taste may vary, but the basic Fate mechanics are very bland, no matter how much you want to dress them up within the fiction. The majority of the crunch comes down to adding +2 to the dice results or rerolling the dice. I don't demand novel dice mechanics for everything, but even Chronicles of Darkness gives things like 9-again and rerolling 1s to spice up the gameplay aspect. (You can't separate the dice mechanics utilized in a game from gameplay. They're integral to gameplay, creating mechanical feedback that influences outcomes, game tone, and mental satisfaction.)
If you're a new GM interested in running Fate, I highly recommend running the game as-written to get a feel for the mechanics, then immediately adding in crunch where you notice the system lacking. I myself have several sticking points that I'll be adjusting for my next Fate game, so stay tuned.
This is going to be a longer series where I talk about the Fate system and my personal thoughts on the system from a traditional RPG player's perspective.