How was the route for the transcontinental railroad chosen?

hoghead has it right on the alternating square miles of land and that land is a huge reason why the remaining RR's are so heavily endowed with money. the real estate business of the RR's is a big business

While all three of the criteria you mentioned were employed to some degree when the Union Pacific and Central Pacific set out to meet and adjoin their two lines (partly because the two had free land to build them) most of the decision to complete them was based purely on speed and completing the railroads as quickly as possible (so building cheap also factored in). While the original line completed at Promontory, Utah has since been abandoned, interestingly, even as quickly as the lines were completed they are mostly still in use today as major rail arteries.

There were a lot of criteria, the mountains had to crossed over the easiest passes but it also had to go through the most profitable areas, which would have meant the richest agricultural areas and established cities. In addition, the first land grant railroads were given adjoining property for each mile of track built so they did go extra distances in some cases. The original railroad barons made a ton of money, not all of it honestly. The benefit to the country was enormous and it enabled the U.S. to settle it's wilderness at a rate that was unprecedented in the history of the world. The raliroads that had to buy their own right of way chose the shortest cheapest route.