What's the easiest way to level up/gain experience in our heroes aside from killing zombies?

Heroes gain Exp from killing zombies, attacking player enemies and also gathering. Upgrading your Heroes Hall will increase the exp earned and there are other buffs from classes and parts that give additional Exp from combat / gathering.

There are also Hero Exp items that you can earn (or buy). These are quite easy to earn from daily challenges and other events. As with the rest of the game earning Exp items gets easier as you level up.