I am unclear on how reequiping gear works. Can a hero requip their gear at the begining of their turn for free but it will cost two MP to reequip if they choose to do so later in their turn? Or does a hero have to reequip only at the begining of their turn and it costs two MP?

The former. Each hero gets a free re-equip at the start of each turn, and can re-equip again mid-turn for a cost of 2 movement points.

Additionally, whenever a hero receives a new item, he has the option to immediately equip it, displacing previously-equipped items into his backpack if necessary.