What group of drugs do tranquilizers and barbiturates belong to?

Collectively, they're Sedatives. However, there's some categorization within the sedative category: Barbiturates, Minor Tranquilizers, and Major Tranquilizers.

Barbiturates (examples)

• amobarbital (Amytal)
• pentobarbital (Nembutal)
• secobarbital (Seconal)
• Phenobarbitol (Luminal)

Tranquilizers (examples)

• Minor Tranquilizers: Benzodiazepines (Xanax, Valium, Klonopin, etc)
• Major Tranquilizers (usually refers to antipsychotics): Haloperidol (amongst others)