I am confused about the usage of double quotes, single quotes, or italics. I know to use double quotes when someone is talking. I know to use single quotes when using quotes within quotes. I know to use italics when using a word in another language or big emphasis. What is the difference between double quotes or italics in a sentence like: She would have to tell them about their "arrangement." She would have to tell them about their arrangement.

Also, she enjoyed being referred to as "Mrs. Johnson." or italics?

Can I use single quotes besides quotes within quotes? If a character is thinking a thought, I have been using italics. For instance:
Let's see him try to put me in my place now, she thought.

Any help is greatly appreciated.

I think double quotation uses to emphasize a word. for example on this case " arrangement" I think we use italic more often than double quotation to emphasize a word.
If you are writing with a specific publisher in mind, you may want to get their style guide before you commit to a particular style.

The common rule is, though, that everything spoken aloud is enclosed in double quotations, while a quotation within a quotation is set apart with single quotes.

You're correct in that italics are used for character thought, and aslo for emphasis. Italicas are also often used for words in a different language from the one mainly used by the characters.

Example: "Yes, I heard her," Joan testified, "she was in the washroom and I heard her say 'I've never been to France in my life!' when Anne asked her if she'd really been in Paris last summer."

The stinking liar, she thought to herself.

"Who lied, mon cherie?"

Sarcasm and double entendres can also make use of the double quotation.

Example: She was "tied up" at the moment.
"My boyfriend", indeed! He was everyone's boyfriend, it seemed.