If you are looking to level your character fast, you will more than likely need to complete as many challenges and milestones as you can. Milestones are similar to the familiar punch card system that Epic Games has used in the past. You are rewarded with 15,000 XP for doing stuff that you would normally do in the game (and some things that you would not normally do). The intent of the milestones is to also introduce you to various aspects of the game to keep it interesting. These milestones can be completed in Solo, Duo, Trios, and/or Squad mode of Battle Royale. It may also be advantageous to complex the milestone in Team Rumble. Unfortunately, there is not a central system to check the status of each of your milestones. I hope that this will be fixed as the season goes on.

Here are the Chapter 2, Season 5 Milestones:

Ignite Opponents with Fire (3, 10, 25, 50, 75)
Ignite Structures with Fire (25, 50, 100, 250, 500)
Consume Apples (10, 25, 50, 100, 250)
Consume Bananas (10, 25, 50, 100, 250)
Use Campfires (3, 15, 50, 100, 150)
Consume Foraged Items (10, 50, 100, 250, 500)
Consume Mushrooms (10, 25, 50, 100, 250)
Catch Fish (3, 15, 50, 125, 250)
Collect Bars (1000, 2500, 10000, 25000, 50000)
Complete Bounties (5, 25, 50, 75, 100)
Complete Common Quests (10, 25, 50, 100, 250)
Complete Uncommon Quests (10, 25, 50, 100, 250)
Complete Rare Quests (10, 25, 50, 100, 200)
Complete Epic Quests (5, 10, 25, 50, 75)
Complete Legendary Quests (3, 10, 20, 40, 80)
Damage Opponents (5000, 25000, 75000, 150000, 500000)
Damage to Vehicles With a Player Inside (250, 1000, 5000, 10000, 20000)
Destroy Shrubs (10, 50, 100, 250, 500)
Destroy Stones (25, 100, 250, 500, 1000)
Destroy Trees (25, 100, 250, 1000, 2500)
Distance Traveled on Foot (1000, 2500, 10000, 250000, 50000)
Eliminations From 150 Meters or More (10, 25, 100, 500, 1000)
Eliminations With Assault Rifles (10, 25, 100, 500, 1000)
Eliminations With Pistols (10, 25, 100, 500, 1000)
Eliminations With Shotguns (10, 25, 100, 500, 1000)
Eliminations With SMGs (10, 25, 100, 500, 1000)
Eliminations With Sniper Rifles (10, 25, 100, 500, 1000)
Eliminations With Common or Uncommon Weapons (2, 10, 25, 75, 125)
Eliminations With Explosives (10, 25, 100, 500, 1000)
Players Eliminated (5, 25, 100, 250, 500)
Distance Traveled Gliding (1000, 2500, 10000, 25000, 50000)
Harvest Stone (2500, 10000, 25000, 100000, 250000)
Melee Eliminations (5, 25, 50, 75, 100)
Sofas, Beds or Chairs Destroyed With a Pickaxe (10, 50, 100, 250, 500)
Melee Damage to Player Structures (500, 2500, 10000, 25000, 50000)
Place Top 10 (10, 25, 100, 200, 300)
Reboot Teammates (2, 5, 10, 25, 50)
Revive Teammates (5, 25, 50, 100, 250)
Ammo Boxes Searched (10, 20, 50, 100, 200)
Chests Searched (10, 50, 500, 1000, 2500)
Supply Drops Searched (5, 10, 25, 50, 100)
Shakedown Opponents (5, 25, 50, 100, 200)
Distance Traveled Swimming (250, 1000, 2500, 10000, 25000)
Upgrade Weapons (5, 10, 25, 50, 100)
Fishing Spot Used (5, 15, 75, 150, 300)
Player Structures Destroyed in a Vehicle (3, 25, 75, 150, 300)
Distance Traveled in a Vehicle (2500, 15000, 50000, 75000, 100000)