Areas where you can begin quests
Diamond City Market [1] – Main quests: Unlikely Valentine, Reunions, Dangerous Minds
Railroad quests: Road to Freedom
Side quests: Painting the Town, Out in Left Field, Confidence Man
Companions quests: Story of the Century, The Gilded Grasshopper, The Disappearing Act
Old North Church [2] – Railroad main quests: Tradecraft, Boston After Dark, Operation Ticonderoga
Railroad optional quests: Butcher's Bill, Mercer Safehouse, Jackpot, Memory Interrupted, Weathervane: Broken Monorail (repeatable quest), Randolph Safehouse (repeatable quest), Variable Removal, Lost Soul, To The Mattresses
Vault 81 [3] – Side quests: Vault 81, Short Stories, Here Kitty Kitty, Hole in the Wall, Dependency
Goodneighbor [4] – Main quests: The Glowing Sea
Side quests: Memory Den, Silver Shroud, Public Knowledge, The Big Dig, Trouble Brewin', Bring hallucinogens for Fred (miscelaous)
In The Third Rail you'll find MacCready - companion quest Long Road Ahead.
Park Street Station [5] – Main quests: Getting a Clue
Pickman Gallery [6] – Side quest: Pickman's Gift
Trinity Tower [7] – Side quest: Curtain Call
Cabot House [8] – Side quests: Emogene Takes a lover, The Secret of Cabot House
Sham. Taphouse [9] – Side quests: Here There Be Monsters
Charles View Amphitheater [10] – Side quests: Pillar of the Commonwealth
Combat Zone [11] – Side quests: Combat Zone