In this paper, we develop the first comprehensive tutorial on designing future 6G networks that synergistically integrate notions of resilience and criticality from the ground up. While resilience refers to the ability to absorb, adapt to, and recover from adversarial or challenging conditions, criticality indicates the degree of importance or urgency assigned to a particular service or component. Despite a spiking interest in designing resilient wireless networks, most prior works do not provide a unified resilience definition, nor harness the intricate interplay between resilience and criticality. In order to fill this gap, in this paper, we highlight the importance of a criticality-aware approach as a key enabler for providing reliable and resilient service functionality. Moreover, we delve into the unique challenges and opportunities of the envisioned 6G features pertaining to resilience and (mixed) criticality. After reviewing resilience definitions, we present a core resilience strategy, a unified resilience metric, different criteria for service criticality, and prioritization frameworks, that augment the 6G resilience prospects. Afterwards, we explore the opportunities presented by promising technologies that enable a resilient 6G network design from a radio access network protocol stack perspective. We briefly revisit state-of-the-art network architectures, establish a rough connection to the Open-RAN Alliance vision, and discuss opportunities, existing techniques, and promising enabling mechanisms for 6G at each layer. Finally, the article discusses important research directions and open problems concerning resilience and criticality in 6G.