Does Armor Stack?
In general, there are two notations for AC:
The most common variety sets your AC to a fixed value, and looks something like this:
AC = 11 + Dex Modifier (Padded armor)
Your AC equals 10 + your Dexterity modifier + your Wisdom modifier (Monk Unarmored Defense class feature)
Alternately, some items increase your existing AC, and look more like this:
Wielding a shield increases your Armor Class by 2 (shield)
... You gain a +1 bonus to AC (Fighter Defense fighting style)
When calculating your AC, you get ONE item of the first kind, and as many of the second kind as you can justify (except where contradicted by rules for specific items, e.g. you can only use one shield).
So if you have a Dragon sorcerer/barbarian, in full plate, with an extra set of padded armor under it, you have to pick which of the four forms of AC you use (Draconic Resilience, Unarmored Defense, Plate armor, or Padded armor). They do not stack.
But if you strap on a shield, that will stack because it grants +2 AC instead of setting your AC to a value. You can even strap on a cloak of resistance for another +1 AC.
What can I wear with Plate?
The rules state that plate armor consists of the following:
A suit of plate includes gauntlets, heavy leather boots, a visored helmet, and thick layers of padding underneath the armor.
There are no rules for substituting one piece of armor for another (e.g. a helm for a crown).
With that said, it is typically traditional to allow replacement of peripheral items (such as gloves, helms, boots, and bracers).