If a character multiclasses into the Barbarian Archetype do they get the Barbarian skills with weapons and armor? Or do they just get rage and athletics?
You only gain what's listed in the Barbarian Dedication feat.
Multiclass isn't at all like it was in PF1.
You become trained in Athletics (or another skill of your choice if you're already trained).
You become trained in Barbarian class DC.
You get an instinct, but not really.
And you get rage.
You don't get weapon or armor proficiency. You don't get all their skills.
In general, the multiclass archetype feats will give you a small smattering of abilities from a given class.
Dedication will give some version of the iconic ability for that class (rage for barbarian, powerful fist for monk, hunt prey for ranger, a few cantrips for casters, etc.), along with 1 to 4 trained proficiencies spread amongst armors, weapons and skills.
After that, you get access to the class feats for your secondary class, and a few feats to improve your basic abilities (2-3 spell slots for casters, ancillary abilities like barbarian instinct, flurry of blows, sneak attack, etc.)