SARS-CoV-2 is a novel coronavirus that causes the respiratory disease known as COVID-19. Despite the urgent need, we still do not fully understand the molecular basis of SARS-CoV-2 pathogenesis. Here, we comprehensively define the interactions between SARS-CoV-2 proteins and human RNAs using Covalent Linkage Affinity Purification, a technique capable of identifying bona fide RNA-protein interactions while reducing deleterious in vitro association artifacts. We show that NSP16 binds to the mRNA recognition domains of the U1 and U2 splicing RNAs and acts to suppress global mRNA splicing upon SARS-CoV-2 infection. We find that NSP1 binds to 18S ribosomal RNA in the mRNA entry channel of the ribosome and leads to global inhibition of mRNA translation upon infection. Finally, we find that NSP8 and NSP9 bind to the 7SL RNA in the Signal Recognition Particle and interfere with protein trafficking upon infection. Disruption of these cellular functions suppresses the interferon response to viral infection. Our results uncover a multipronged strategy utilized by SARS-CoV-2 to antagonize essential cellular processes to suppress host defenses.