Say for example I was wearing Haramaki and cast Mage armor. Is the armor bonus now +5? Is this different for shield? Thanks.
Unfortunately not. Since the Haramaki and Mage Armor both grant armor bonuses to AC, they do not stack, so you would get the higher of the two bonuses (the +4 from Mage Armor) and the Haramaki would have no effect while the Mage Armor is up. Note that, if you enchant the Haramaki with special properties (eg. +1 Light Fortification Haramaki) the special properties will still have an effect, although the +1 Harakami armor bonus is lost as it is less than that of the Mage Armor.
Importantly, enhancing your armor gives an enhancement bonus to the armour, rather than to your AC, so a +1 Haramaki with Mage Armor ignores both the Haramaki base armor and the +1 bonus. However, if your armor bonus is greater than the +4 from Mage Armor, you will get it as usual.
As previously stated, the bonus from Mage Armour and the bonus from your Haramaki would overlap and not stack, the same applies to Shield and any other shield bonus to AC (such as from carrying a shield).
In Pathfinder, bonuses of the same type do not stack; you just get the benefit of the highest bonus. This rule has a couple of exceptions (such as dodge bonuses) but mostly holds true universally. Penalties, on the other hand, always stack; 'cause yolo.