Leveling is not a very important aspect of the game. Through campaigns and event quests you will level your account fast. The materials needed to get your toons strong will take time. Increasing the level of your toons' Ranks and Tune-ups will have thew biggest impact on your team.

Marvin's Invasion
Marvin's Invasion gives you items that will help with the Region you are working on. For example, if you are working the Space quests in the invasion you will get toons and leveling materials for the toons that are in the space region. Use energy for the Invasion if there aren't any good things to use your energy on in the Events Board. 

​Gadget Campaign
Like it says, this campaign is designed to provide you with gadgets for your toons. Eventually you will destroy most of your gadgets to make parts for leveling the ones you use. 

Daily Quests
This is your first priority. The quests are seriously buggy while you are leveling.

Punch In
One click and you get some really good prizes.

Arena
Do all three arenas every day. It doesn't really matter if you win or lose, just get the stones.

Tower
The tower gives you some character pieces and more important, Portal currency. These can be used to purchase some much needed character pieces in the the toon store.

Brawl Banks
Try to fill up your bank with gold chests before you log off.

Energy
Use all energy if possible. Repeat quests to help get needed items.

Star Crates
Ideally, try to clear three crates. You get a lot of really good items from the Star Crates.