During interphase of the cell cycle, the key event that occurs is the replication of the DNA.
Interphase consists of three phases: G1 (Gap 1), S (Synthesis), and G2 (Gap 2). DNA replication occurs specifically during the S phase, when the cell copies its DNA in preparation for cell division.
The other events mentioned—spindle formation, condensation of the chromosomes, and separation of the spindle poles—occur during mitosis or meiosis, not interphase.