In films, documentaries, and TV shows, that focus on the history of the electric telegraph, they often discuss the trials and failures of laying trans-Atlantic cable, or how Samuel Morse came up with his code, and so on, but there seems to be no information on the engineering/logistics of how one actually sent a message. Today you'd need to dial a certain phone number or type in a particular email address to make sure your message was received by the intended recipient. I've never seen any depictions of "dialing" a telegraph; in the films they just start tapping away on their morse key. How did they know they were sending their message to the correct destination?
In brief, the first few characters were the "addressee," alerting whoever should listen up.
Think about a single radio frequency that various people with walkie-talkies can transmit on and receive. So the first thing a speaker trying to reach "George" will say is "George; come in, George." George perks up and answers "George here. Go ahead, Tom." Same with telegraph operators along a circuit in, for example, the different stations and control towers of a railway line. Dozens of telegraphers might be on that same circuit. The first characters transmitted would typically be the two-character code of the station for which the message was intended. That station would respond with a short code indicating it was listening and copying down the message. Other short codes known as wire signals could ask the sender to repeat a section, indicate that the message was finished, and similar things. When things were busy, it might be a couple of minutes before the wire got quiet and you could send your message. In the wee hours of the morning, telegraphers would chat about the weather or whatever just to stay awake.
While it's practical for all the stations on a railroad division to share one circuit, obviously that's impractical for a national network. But even for nationwide service via a company like Western Union, the first few characters would indicate what city or branch office the message was intended for. That would alert the repeating stations that needed to retransmit the message, and finally the receiving station where the message would be written out and delivered. Also within a telegram would be the recipient's address, though that might be as simple as "PALADIN, SAN FRANCISCO."
By the 1900s, such "common-carrier" telegram services had fairly sophisticated switching that could direct thousands of messages each day where they needed to go without burdening irrelevant offices, though railroad telegraphy never changed all that much. By the 1920s, commercial telegrams typically were sent by automated teleprinters that were switched to the proper receiving office in much the same way as telephone calls.
You would walk into a telegaph station. An operator would take your message- either you could dictate or you could write it. He ( and operators seem to have been mostly male, and often young) would go over to a telegraph key and open it, which would break the connection through the key and make it possible to transmit ( the origin of the phrase "open a line") . Every station had a two-letter ID "sine", and the operator would send the sine of the receiver twice, his own station's sine once, until he got an acknowledgement " II" ( dot dot... dot dot) and the sine of the receiving office, and the signal to go ahead ( dash dot dash) ( or until he decided there was something wrong with his equipment or the line, in which case he would start investigating). After hearing "go ahead" ( dash dot dash) he would signal "starting message" ( dash dot dash dot dash). Then there would be something of a dance between sender and receiver, trying to be efficient and quick, but also making sure there were no errors: either could signal they had been interrupted, needed to repeat something, could tell the other to wait because someone was at the counter , etc. And another station might interrupt with another message ( and the other operators could demand to know what was so important, anyway).
The message might be full of abbreviations, special terms and shorthand codes. The railroads were very reliant on telegraphed signals, had their own operators working with their own codes in their own format for sending orders and messages about trains. But businesses could also develop special lists: a Ford dealer in Biloxi could send a telegram of only several words to Ford in Detroit that would still. be able to specify the needed parts for a Model T as well as specifying how they were to be sent and the terms of payment.
When the sending operator had finished, he would signal end of message (dot dash dot dash dot) and the receiving operator would reply with "OK"(dot dash dot). If he was sending another message to the same place, he would signal with "attention" again (dash dot dash dot dash). If he was done , he'd signal transmission finished (dot dot dot dash dot dash).
Operators would gain a fluency with experience, and could often identify each other by their "fist", or way of sending code. But they also had their own ID letters, which could be asked for.
G.M. Dodge : The Telegraph Instructor