In my parish, the lector coordinator is always “scouting” for readers. If she gets to know a parishiner who is reliable and responsible she asks them if they are willing to be a lector. If the person agrees, she meets with them for about an hour and trains them about how to properly conduct the readings during Mass. She then has them “practice” during weekday Masses and then the new lector is placed on the lector schedule for Sunday Masses.
If you are interested in becoming a lector, I’m sure it’s just a matter of letting the Lector coordinator, or whomever has this role in your parish, know. I’m sure they will be excited to have a new lector!