Railroad's hideout is very well hidden and it is under the Old North Church in the central part of Boston. You find the hideout during Road to Freedom, where you find hints on how to get to the HQ, and where you meet with Desdemona, and other important faction members, for the first time. You can start this quest at any moment of the game (it appears in your journal after you eavesdrop on the conversation in Diamond City). If you ignore this quest, you will not miss the opportunity to meet with the Railroad, because you meet the members of that faction during an "independent" quest - The Molecular Level. After you complete Road to Freedom, you can take on quests given to you by The Railroad. This both goes for main quests (Tradecraft and the following ones) and for side quests (many of them are random).
While completing the abovementioned The Molecular Level, you can (although this is optional) ask The Railroad for their help at building the teleporter. When this happens, Desdemona tasks you with Underground, Undercover. Completing this quest is synonymous with tightening your ties with this faction and spoiling good relations with the Institute and The Brotherhood of Steel. The last main quest for this faction is The Nuclear Option and the objective here is to destroy the Institute HQ.
While working for Railroad, you can receive Burning Cover - this quest becomes available if, as a result of your actions, you become the enemy of the Institute (e.g. as a result of murdering a member of that faction). This works both ways, because you can become the enemy of Railroad if you accept specific quests from other factions. The ones in question are End of the Line (the Institute) and Blind Betrayal (the Brotherhood of Steel).