The reference you're looking for is on page 197 of the Player's Handbook, under, naturally enough, Damage Resistance and Vulnerability. Specifically the third paragraph states:
Multiple instances of resistance or vulnerability that affect the same damage type count as only one instance.
So the Bear Totem Barbarian still takes half damage from bludgeoning, slashing, or piercing attacks.
Damage from the same source with multiple damage types (such as a grimlock's Spiked Bone Club attack, or the flame strike spell) is all inflicted at once. Nothing in the rules for Damage Types (PHB, p. 196) indicates that a given effect is restricted to a single type of damage. The wording for these effects doesn't separate the damage types into different sources:
Spiked Bone Club, MM p. 175
Hit: 5 (1d4 + 3) bludgeoning damage plus 2 (1d4) piercing damage.
Flame Strike, PHB p. 242
A creature takes 4d6 fire damage and 4d6 radiant damage on a failed save ...
If flame strike's damage types were supposed to be delivered by different sources, then they wouldn't be conditional on the same (singular) saving throw. Likewise, the Spiked Bone Club's damage is the effect of the (singular) Hit.
To further support this, the optional Vitality rules (presented in this Unearthed Arcana article) use this language:
Whenever a character takes 10 or more damage from an attack or effect, the character loses vitality.
This makes it clear that a single attack's total damage, regardless of its constituent types, is what matters. The Massive Damage and Instant Death rules have the same flavor and can be construed to work in the same way.
Of course, resistances and vulnerabilities may apply separately to each damage type that an attack inflicts. A monster that is resistant to fire but vulnerable to radiant damage that fails its save versus a flame strike will take an average of (14 / 2) + (14 * 2) = 35 damage from the spell as a single damage source.
While it doesn't seem to be explicitly spelled out anywhere, the Sneak Attack entry says:
This is extra damage dealt by the weapon attack; therefore, it should deal damage of the same type as the weapon. As backup for this, the section on critical hits says that:
It seems clear that Sneak Attack is extra damage added to the weapon attack, and therefore has the same damage type as the weapon.
Specifically, the Spy uses a shortsword, which deals piercing damage, so his Sneak Attack damage will be piercing also.