Daily Summary

Here are more details on each of the major sources of XP:

Quests are the main way players will earn free in-game cosmetics and XP.
The following are only available in the Team Rumble and Battle Royale game modes (Solo, Duos, Trios, Squads):

• Daily Quests
  Players should see three Daily Quests available to them. Once they are completed, they should refill up to three available for your next round.
  Squadmates can complete (and also earn XP from) the active Daily quests of their teammates.
  Each Daily Quest is currently worth 1000 XP.
  There is also Daily Bonus Goal, which is awarded for the first three daily quests completed each day. They are worth 3x 15,000 XP.
  Daily Bonus Goals reset at 9am EST each day.
• Weekly Quests
  Weekly Quests are typically available until the end of the season, and are released on Thursdays.
  Each Weekly Quest is worth anywhere from 15,000 - 40,000 XP.
  There are Weekly Bonus Goals which involve completing a certain number of Weekly Quests. Once they are all completed you advance to the next Tier of Goals.
  Tier 1 Weekly Bonus Goals are worth 40,000 XP each.
• Milestone Quests
  Milestone Quests last for the the entire season, and reward you for completing in-game activities.
  Each Milestone Quest is worth 2,000 XP.
  There are Milestone Bonus Goals which involve completing a certain number of Milestone Quests. Once they are all completed you advance to the next Tier of Goals.
  Tier 1 Milestone Bonus Goals are worth 3x 30,000 XP each.
  Honestly, I wouldn't focus all that much on Milestone Quests beyond earning them as you go. Some of them require a ludicrous amount of effort to achieve with intention, for relatively inconsequential XP. They should be viewed as tasks you complete only when they become available during general gameplay.
• Other Snapshot, Character and Story Quests
  Other thematic quests can appear weekly, and offer either XP or cosmetics rewards. They may have an availability period before expiring, or more commonly last until the end of the season.
  XP rewards for these quests are typically in the 20-40k XP range.

The Save The World game mode requires players to have bought it separately from the base game which is free to play.
That being said, you can earn XP in Save The World mode which is counted toward Battle Royale and the Battle Pass. There is a daily limit in the range of 450k XP, so this is worth looking into.
Ride The Lightning can pull in over 30k XP per round.
It is possible to achieve AFK farming here if you are sufficiently far along ingame, using the Endurance mode.

Creative Mode maps can feature "accolades", in-game events or achievements which reward players with XP. These range from completing obstacle courses, to opponent kills, to simply interacting with hidden map features.
This is lower in the list because the XP awarded varies wildly. You can find videos online advertising "creative xp glitches", but they can be very inconsistent. The game adjusts XP awarded dynamically, based on how fast you're earning, to discourage abusing the mechanic.
Newer "UEFN" maps created using the newer unreal engine framework will also grant XP for simply being in-game, although the amounts earned do not appear to be consistent. You will usually get the XP awarded retroactively some tens of minutes (or even hours) after finishing a play session.
Creative XP amounts are capped each day, and the amounts earned will scale down over time. Some accolades will pause earning if the game detects you have not moved around in awhile.

Opening chests, ammo boxes, freezers, safes, llamas, drops award XP.
Collecting foraged items awards XP.
Interacting with NPCs awards XP.
Eliminating players and other in-game activities will all assign XP.

Enjoy the game and the XP will follow :)

This doesn't award extra XP so much as it catches you up on a little bit of the XP you missed by not playing the day before.

If you didn't earn at least (used to be Daily Quest amount) from the day before you may notice your XP bar glowing gold and labelled SUPERCHARGED XP. The idea is that supercharged status grants a 5x multiplier on any XP earned, up until you earn (what used to be 30,000) XP. It's not clear what the current Supercharged XP limit is but it is likely below what could be earned doing the Daily Quests from the day before.

It's a way for you to skip a day or more of Eat-Sleep-Fortnite (seriously, go outside) and still keep up a bit of your leveling pace.

No longer available in-game.

This is no longer a thing in Battle Royale.