Do NOT spend Gems (those blue gems that allow you to skip things) yet. These are incredibly valuable and the game makes it seem like you can just throw them away. DO NOT WASTE THEM. You need to save 300 of them to summon heroes. NEVER EVER use them to do a daily summon. ONLY use them to get guaranteed 3 stars or higher.
Do NOT train with heroes you want to keep. If you use a guy to train, that guy disappears forever. So there are two types of heroes: those you upgrade, and those you sacrifice to those you upgrade. You NEED smaller guys in the future, so once you upgrade someone, do NOT sacrifice them away! Also, I suggest saving special 'Trainer' heroes to use when you get 4- or 5-star heroes. You WILL get these in about a week or so (depending on how much you play), so BE PATIENT!
Do NOT level your one star guys! Except maybe the one they force you to ascend during the tutorial.
Do NOT train with heroes of a different color. You get far better benefits from similar color guys.