For instance, if I was an American businessman planning a trip to Italy would I have to physically bring all of the money I wanted to spend with me? Additionally, what measures were put in place to avoid fraud?
I answered a very similar question here
In your specific case of traveling to Italy from the US pre-1800, I am not sure. But, generally,, you would look to find and pay a merchant or bank that did enough business there to make up a letter of credit that said, essentially, pay John Smith 1,000 ducats. When you arrived, you would go to that other merchant or bank and present your letter. There would be signatures on file and on the document to be compared, that could discourage fraud. There would be a discount- i.e. they might only pay you , say, 950 ducats. There might also be a delay: they might hold the note and ask for someone to vouch for you. You might also get a couple of letters, for different banks, etc., as you might not know until you got there really whether a note/letter was good.