I’m quoting this here and adding a bit more to make this advice actually somehwat actionable instead of just vague. You can, and probably want to, embed setting information right into character creation, and there are a number of ways to do this.
Dungeons and Dragons tacitly does this, and the best approaches to DnD are the ones that acknowledge that there’s an implied setting that arises from minor things like choosing your alignment, picking what class you will play as (and the list of classes available to play as). This gets a bit more interesting not too long after DnD first came out, when Traveller made character creation a little mini-game of rolling dice through your career path until your character died or mustered out. Runequest was a classless, skill-based system that had way too many innovations to list, but some of the ones applicable here include its very pragmatic approach to spell-casting, and the way magic was tied in to membership in mystery cults. It was even the source for modern D&D’s prestige classes, wherein once your character passed a certain threshold and did certain things in game, they could become a ‘rune lord’.
So, there’s a brief overview of where this approach to mechanics is coming from. Now on to some modern and semi-modern examples.
Talislanta was basically a 90s D&D wannabe, but it ended up being pretty influential on quite a few indie game designers. Character creation was ludicrously simple: pick an archetype you liked, move a few numbers around, and you now have a character. Feng Shui also had the same idea, also in the 90s. The key way this differs from class-based systems was in the quantity and diversity of archetypes. Feng Shui had every single action movie stereotype you can imagine, and succeeded at the goal of emulating its milieu. Talislanta, though, traded on the depth and cultural diversity of its setting, so it had hundreds of archetypes, many of them culturally coded. By looking through the choices available, a player can already piece together an idea of what each of those cultures was like. And you don’t have to do a half hour spiel about the setting or require players to do anything they wouldn’t already do.
Dogs in the Vineyard is a western rpg about wandering quasi-mormons solving problems and dealing with the fallout of their choices. It has the usual ‘explanation of setting’ section of the rulebook, but it does something clever in character creation. The last step of character creation is the initiation of the characters into the holy order all characters are members of. Unlike most games, this is actually role-played out and serves as a prologue for your character. It teaches the setting, teaches the mechanics (because you do your first dice rolls within this initiation) and ties your character more fully into the setting.
Burning Wheel is a complex and heavy game that feels like generic fantasy if you glance through it, but implies much about the setting beyond what you would first expect. It does this in a few ways, but the main two are its lifepath system, and its natures. The lifepath system is more abstracted than the one from Traveller, but it still encourages thinking of one’s character as a collection of past experiences. Each lifepath is made up of perhaps some stat adjustments, a collection of skills you can spend points on (the first one listed getting a free boost), and a collection of ‘traits’, little descriptors that give you a bonus if you roleplay them, some of them having tiny mechanics attached in addition to their generic use. Finally, there are the Natures. I’m not sure if they work this way in BW, since I mainly played mouse guard and torchbearer, but the nature is a sort of amalgamation of all the qualities that makes a dwarf dwarflike, or an elf, elflike, or a human like a human. These things are described in the book and very quickly explain what makes those fantasy races unique in the implied setting. This nature has a numerical rating and it can be added to your roll if you are engaging in an action that fits your nature, or it can be added to your roll if you do an action that goes against your nature but the rating is reduced after the roll. This obviously encourages players to roleplay their characters without forcing them to stick to the broad archetype.
Finally, there’s Beyond the Wall, an OSR D&D clone with a frankly amazing character creation and setting creation system. To make a character, you fill out a questionnaire attached to the character archetype you want to play, this questionnaire asks backstory questions (and this is where a lot of setting gets embedded), questions about your relationships to other characters, etc. These both build the character and define the character’s place in the setting. It’s less bookwork than the lifepath/careerpath systems but with significantly more diversity of characters than the archetype/class systems.
Ok, I have to head out the door but I wanted to provide some food for thought on ways to actually tie setting and mechanics together.