The fastest way to level up is to play Evacuation. At the end of the 5 matches, you'll get a substantial XP bonus based on your performance.
If you're looking to get this as quickly as possible, set up a solo custom match and choose Evacuation. You can get through a campaign as the Monster faster, but choose whichever side you're most comfortable with. Set the difficulty to favour whichever side you're playing.
This is also a good method for boosting mastery and for getting the game mode achievements for Nest, Rescue and Defend.