The giant $k$-core --- maximal connected subgraph of a network where each node has at least $k$ neighbors --- is important in the study of phase transitions and in applications of network theory. Unlike Erdős-Rényi graphs and other random networks where $k$-cores emerge discontinuously for $k\ge 3$, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We also develop a $k$-core calculation that includes clustering and provides insights into how high-level connectivity emerges.