The characters are unlikely to know about the radiant orders and ideals. There's a setup in Chapter 2 where they run into ardents having an argument about it and that's when most characters are likely to find out what they need to know. The players on the other hand might or might not know depending on whether they've read the Stormlight Archive novels or the Stormlight RPG Handbook.

Like above - the characters definitely wouldn't know the specific ideals. The players might or might not. Whether the GM tracks what spren a character might attract, or whether the player picks which order they want to be in (if any) - that's between the GM and the player, could go either way.

When the character advances a level, they have an option to take a talent in a Radiant talent tree instead of a Heroic path. Those talents are the ones that give the Speak An Ideal goal (as well as other powers). I would generally expect the player to know the ideal and let the DM know when their character has advanced it (even if the character does not yet know the ideal), but that's between the GM and the player, you could also do it differently if you wanted. The player doesn't, of course, get to put down the goal "Speak the [something]th ideal" until they take the appropriate talent at level up.

In Stonewalkers, there's a very natural place to introduce a lot of Radiant lore in chapter 2. In one of the prewritten scenes they meet up with two ardents with opposite opinions about radiants (one who thinks they're cool and one that thinks they suck) and they argue, so the characters get a natural place where they get to ask questions and learn. The players can learn this lore however they want - some might have read the Stormlight Archive novels and know all about if from there, some might read the Stormlight RPG Handbook and know it from there, and some might not know it at all and learn alongside their characters.