I didn't notice anything in the rulebook about how many items can be used in each player's Hero Phase. Is it just 1, or are multiple items allowed to be used?
You can use as many items as you like, as long as you follow the rules on the card. For example, if you have two Potion of Healings you can use them both during your Hero Phase, but not during another player's turn. You may use a Heavy Cloak on the Villain Phase during your turn, or another player's turn. A Flask of Oil is used instead of attacking, so you can only use one on your turn.
From the rulebook:
You can benefit from multiple Treasure cards in play that apply to your Hero. For example, you can use both a Fortune and an Item to increase the bonus to an attack roll.
"Multiple Treasure cards" includes using multiple Items (subject to the rules listed on them). However,
The only exception to this rule applies to Items that provide attack or defense bonuses. A Hero can gain only one attack bonus and one defense bonus from Items at a time.
This exception means that you cannot use two Belt of Storm Giant Strengths at the same time.