Suppose it’s 1935 and I’m in Miami Beach. I want to send one telegram to someone in Chicago, and another to someone in New York City. By what mechanism are the telegraphs routed to the appropriate city? Be brief, or write a dissertation. 
Thank you.
This seems to be a popular topic lately, anyone know what is driving the interest?
Here is a recent thread that speaks to your question. Here is another one. Feel free to ask for further specifics.
Get yourself down to the bookstore or the library and get a copy of “The Victorian Internet”, by Tom Standage. Great book that will give you just about all the information you could want about the development, building, and operation of the telegraph system.
Yes but that is the mid 18th century and it was probably a lot easier by 1935. But it is a fascinating book.
This is only a guess based on my knowledge of telegraphy and the time the OP has mentioned.
By 1935, most main line telegraph stations would be using tape or page teleprinters for telegrams. As an example, the Creed model 7 was introduced in about 1931. The machines can be connected point-to-point or via a switching system, controlled by an operator. They simply send to the appropriate terminal station which can then relay if necessary to the closest station to the recipient. I don’t know anything about the specifics of the US networks, so can’t say if the first section would be direct to New York/Chicago or would be relayed through somewhere else.
Imagine the network working something like an airline’s network. There will be direct lines for high traffic routes and hubs for collecting from lower traffic areas.
Yep, this is the source of the pasted-up telegrams you see in movies.
In one respect, though, by 1935 things hadn’t changed that much from the Victorian age. Long distance calls were still horrendously expensive so telegrams were still the preferred way of sharing personal news and other information with family and friends. In The Caine Mutiny–the novel not the film–Willie calls his girlfriend from Hawaii to New York, and it costs eleven dollars and change. Given the fact that this would be like $150 or $200 today, Wouk obviously put that in to remind us that the protagonist comes from a lot of money.
Most cities had Western Union offices. You would go down to the office in Miami Beach, fill out a form, and have them send the telegram. The Chicago office would deliver the pasted up telegram to the local address.
The telegram system involved wires connecting to cities, so Western Union would rout the telegram to a particular city. The Miami office might have a direct line to Chicago, but if not, they’d send it to a city that did. In the old days, there was a separate line for each destination; if you had to go through Atlanta (to make up a route for this example) to get to your final destination, the message would be resent from Atlanta to Chicago.
I remember receiving birthday telegrams when I was a kid in the 50s. By that point, Western Union would call and then ask if you want delivery (for a higher fee).