Need to replace hard drive. How do I install OS after?

My Mac Mini will not boot normally. Was able to select recovery disc after holding down option key during startup. After starting disc utility it became clear my "hard drive has damage that could not be repaired". So I watched a bunch of videos about replacing hard drive and I am very confident about doing it myself. The problem I see is after installing new HD, how do I install OS? I mean I have a current backup on an external, but does that have everything I need on it? Just did standard Time Machine backup.