I cannot figure out whether you get proficiency in the Skills/Saving Throws of the new class (i.e. I have a 1st-level Barbarian and went into Druid from 2nd level onwards--do I receive the Druid's proficiency in Intelligence/Wisdom saves and the option to choose 2 Druid skills in addition to the Saves and Skills I have as a Barbarian?
No. The multiclassing rules address this point specifically:
When you gain a level in a class other than your first, you gain only some of that class’s starting proficiencies, as shown in the Multiclassing Proficiencies table.
To answer your specific example, when you multiclass into Druid, the only proficiencies you gain are with light armor, medium armor, and shields. All of which you already have from Barbarian.
It's noteworthy that no class grants saving throw proficiencies when multiclassing, and only Bards, Rangers, and Rogues grant any skill proficiencies.