If you're looking to level your characters up quickly, the best thing you can do is to visit the Quest Board in your Backyard Battleground.
Pick up quests to complete, and get them done. When you complete a quest, you'll increase your experience multiplier, which means even quicker advancement. There's a ton of Quests, too, including some specifically for the Backyard Battleground and for multiplayer. No matter what area of the game you feel like playing at any given time, you'll be able to complete quests progress.