During conversation, people often endeavor to convey information in an understandable way (finding common ground) while also sharing novel or surprising information (exploring new ground). Here, we test how friends and strangers balance these two strategies to connect with each other. Using fMRI hyperscanning, we measure a preference for common ground as convergence over time and exploring new ground as divergence over time by tracking dyads' neural and linguistic trajectories over the course of semi-structured intimacy-building conversations. In our study, 60 dyads (30 friend dyads) engaged in a real-time conversation with discrete prompts and demarcated turns. Our analyses reveal that friends diverge neurally and linguistically: their neural patterns become more dissimilar over time and they explore more diverse topics. In contrast, strangers converge: neural patterns and language become more similar over time. The more a conversation between strangers resembles the exploratory conversations of friends, the more they enjoy it. Our results highlight exploring new ground as a strategy for a successful conversation.