As the triggered ability resolves, check which creatures on the battlefield have any of the creature types that the previously enchanted creature had as it left the battlefield. If that creature was a Goblin Rogue, for example, Reins of the Vinesteed can be returned to the battlefield attached to either a Goblin creature or a Rogue creature.

This triggered ability doesn't target a creature. You may return Reins of the Vinesteed to the battlefield attached to a creature that has shroud, for example. However, the creature must be able to be legally enchanted by Reins of the Vinesteed. You can't return Reins of the Vinesteed to the battlefield attached to a creature that has protection from green, for example.

Reins of the Vinesteed's last ability triggers if it and the creature it's enchanting are both put into the graveyard at the same time, or if the creature it's enchanting is put into the graveyard but Reins of the Vinesteed isn't. (In the second case, Reins of the Vinesteed is then put into the graveyard as a state-based action.)

You can choose to return Reins of the Vinesteed to the battlefield attached to a creature another player controls. You still control Reins of the Vinesteed. When that creature is put into a graveyard, the ability of Reins of the Vinesteed will trigger again and you'll be able choose another creature for the Aura to enchant.