By strict RAW, you can sense the presence of illusions of all kinds, but you can't actually see exactly where they are, or that they are illusions rather than just some kind of magic in your vicinity. From the description of Detect Magic:
For the duration, you sense the presence of magic within 30 feet of you.
Simple enough; if there's magic, you can sense it.
If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
If a creature or object is covered by an illusion, they're not a visible creature or object, and the illusion itself is neither a creature nor an object, so you can't see its aura. An illusion that is not being "borne" by a creature or object is likewise not a visible object or creature, so you can't see its aura, either.
Where things get messy is if there is a creature or object that "bears" an illusion but is still visible. However, I can't find any cases that would fall under this category. For example, you could use Minor Illusion to cover someone's head, but that is a freestanding illusion that happens to be around a creature, rather than actually being borne by that creature.
As far as whether you want Detect Magic to function by strict RAW in your game is concerned, given the way you've phrased this question I assume you're more worried about it being too powerful than the opposite, so this should suit you.
Magic awareness just states that "Until the end of your next turn, you know the location of any spell or magic item within 60 feet of you that isn’t behind total cover. When you sense a spell, you learn which school of magic it belongs to."
According to this I would think, that a Barb would just immediately be able to Discern Illusions like Major Image and know the location of Creatures with greater Invisibility. Right?
– DungeonKnob1321 Jun 08 '22 at 20:48