What process was involved in transferring money in 1875 from the East
Coast to someone in, for example, Wyoming? How was the actual cash
"delivered" and was it transferred from bank-to-bank or bank to post
office?

Hello and thank you for your question.

Famous for its pioneering telegraph service, the original Western
Union dates back to 1851 and introduced electronic money transfer
service in 1871.  The Western Union Money TransferŽ service soon
became the company's primary business.

About Us - - Our History
http://www.westernunion.com/info/aboutUsHistory.asp

http://www.cnet.com/investor/news/newsitem/0-9900-1028-20149218-0.html

http://www.wu.co.yu/faq.html

http://www.westernunion.cz/en/start.php

http://www.e-moneytransfer.it/chi_eng.htm

So all it took in 1875 was to visit an east coast branch of Western
Union, deposit the funds, and the recipient could pick up the funds at
the local branch.  Of course if (presumably) there was a greater
volume of cash being wired West compared to East, then periodically
the Western Union folks would have to make up the shortage by
delivering funds to the western branches.

Search terms used:
 history "western union" "money transfer
 1871 "Western Union Money Transfer" service 

Thanks again for letting us help with your search.

Sincerely,
Google Answers Researcher
Richard-ga