It's 1850 miles from Reno to St Louis is a straight line. On a good day on good surfaces (I won't call them roads) a stagecoach could cover 70 miles. Thus the journey would take, at best possible speeds, 24 days. I think it's fair to say that the journey would really take at least a month, since you have to get over the Rockies and that's going to slow the average speed. You've really got to want to be in Reno to put yourself through that.
Today the journey can be done in 3-4 days by car.
At the time of this stagecoach St. Louis would be just about the westernmost point of "civilisation", Denver a far flung Mountain town, and Reno? Not the pocket Vegas it is today, but a trading post for trappers, loggers and gold seekers.