Start the game and first go talk to the Old Miner and ask about his hat. Then go to the Train Station and talk to the ticket salesman and ask about a specific ticket. Now talk to the Station Master and get the mission to clear the tracks. Return to the Old Miner and get his hat.
You can either go directly to the obstruction and shoot all of the Beefhead gang, including the ones in the room up the ladder. You can also head to the Ghost Miner and walk with him to the Beefhead gang to clear the tracks (the ghost miner will scare them away).
Go back to the Train Station and talk to the Station Master. Now you need to find three hands to help clear the tracks of debris. Three guaranteed helpers are Rancher Richard (Southeast of Clintville) and Rancher Bell and the Old Miner (both near each other in Clintville). Go talk to each one and then go straight into the tunnels to meet the Station Master and the three hands. Return to Clintville Station and talk to the Station Master once more.
While there, talk to the Ticket Salesman and ask about a timetable. When he asks, tell him you are from Transcontinental Express.
Go back to the Station Master again and then you have to wait for the Train to show up with your sister.
The quickest way to make the train show up is to just head west on the tracks until you get to Santa Anna Station. Head North to the Haberdasher and ask her about the business and then offer to find her hands to run the store. Now go out and talk to all of the women in Santa Anna and Santa Anna Station until you find three that you can ask to help, then return to the Haberdasher.
Once you fast travel back to Clintville Station and talk to your sister, the achievement will unlock.