Game mechanics are an extremely important segment of gameplay because they directly affect the control complexity and in turn the learning curve. For a simple game with limited mechanics (say an arcade racer with only steer, throttle and brake) a designer need not layer in the mechanics. But, for a more complex game (Halo, Splinter Cell, Zelda, Ratchet & Clank), mechanics progression is essential so as not to overwhelm the player with the complexity of the controls.
The 2 main styles of mechanics progression that can be used in games with even a moderate level of mechanics complexity are what I call Gated Access and Directed Gameplay:
Some games use one or the other and some use both (Halo for instance requires more skill with the mechanics later on and opens new weapons and vehicles to be acquired / used).
Weapons or any inventoried mechanics are an obvious segment to reveal progressively via Gated Access. The best combat games actually reveal a new weapon sooner than they make that weapon consistently available to the player (for example an enemy may have a new weapon, and you can kill them to get it, but only with a limited supply of ammo up front). Deeper driving games should manage the progression of mechanics by directing the gameplay to control when more complex mechanics must be used and at what frequency (see Fig 1).
Platform games like Ratchet & Clank or adventure games like Zelda have entirely separate levels dedicated to new mechanics and these top-quality games take a lot of care to reveal these mechanics progressively, and they feature them even more by making all content on those levels support the new mechanics (for example this leaf-wielding flight and fan level in Zelda: The Wind Waker). Most great RTS games (Command & Conquer, etc.) control the progression during the single player campaign by initially limiting the tech level drastically at the beginning so that you learn the basics before more buildings and units are revealed at later levels. Many great action games also reveal new attack moves, or abilities as the game progresses.
The thing to remember is that most any game in any genre (maybe other than team sports) can and should have their mechanics structured as part of the gameplay progression in order to deliver a great experience for the player (easy to grasp, keeping the user wanting more, and rewarding continuation with new abilities and/or new challenges).