I visualize that expensive items were literally paid for in cash and people going to Europe, for example, wanting to spend the season in style, brought stacks of thousand and five thousand dollar bills with them. Maybe there were letters of credit or other instruments your bank could prepare for you to present at a European bank but that seems cumbersome. On the other hand, carrying so much cash seems dangerous. Personally, I never saw my parents with much cash but my grandfather (born at the beginning of the 20th) did tend to carry a wad.
Large sums of cash were actually the instrument of choice in many applications. It seems strange and unsafe today, but there was usually no alternative. Depending on the amount involved you very well might hire a team of guards to safeguard you and your money as you moved, and you would be keen on taking advantage of things like hotel or train safes so you could avoid carrying it on your person as much as possible. If you were transporting a truly staggering amount of money you might very well accomplish it by filling up a safe and then shipping the entire safe between locations.
One duty that involved shipping and handling large sums of cash was payroll- paying the workers of a company. Each worker had to be paid in cash, so this meant carrying enough cash from the bank to the factory or job site to pay every worker their (typically) weekly or bi-weekly wages. Payroll was thus a dangerous job to do. Whether the money was carried, driven via a horse/coach, or moved on the train, those large sums of cash were favored targets of bandits and criminals.
Payroll was commonly transported via train where practical. There was a special car on trains called the "express car" whose job was to transport highly valuable articles such as cash or gold, and the people who guarded these cars were called "expressmen". Many of the daring train robberies you read about were targeted at these specific cars- some heists actually succeeded in detaching and stealing the entire car, leaving the robbers plenty of time to loot it of valuables.