This quest is triggered after concluding Jewel of the Commonwealth. Do this by visiting the Valentine Detective Agency in Diamond City. After agreeing to find Nick Valentine, head to Park Street Station, his last known location. Take your time preparing and arming yourself. While Park Street Station isn't far from Diamond City, it is dense with hostiles.
As a quest that takes you through a linear subway and a vault, this rescue mission is straightforward. The challenge is in taking on the dozens of 1940s-inspired gangsters along the path. Also note that the subway and vault has two collectibles we recommend you pick up: the Speech Bobblehead and an issue of Astoundingly Awesome Tales.
After rescuing Nick Valentine, make your way out of the vault and subway. The biggest hurdle of the quest is Skinny Malone, his dame Darla, and two bodyguards. With your companion of choice and Nick helping out, Malone and his gang shouldn't be a huge hassle. Just be crafty with your surroundings and use the boxes and crates for flanking opportunities.
With Malone out of the way, the rest of the journey is smooth sailing. Meet with Nick at his office in Diamond City to trigger the next main quest.