Use quotation marks or italics for written quotes?

Normal style in narrative writing is to use quotes, not italics.

Some authors of fiction, notably James Joyce and William Gaddis, used an em dash at the beginning of a paragraph that started with a quote:

— Do you have the money? he asked.

But that was an idiosyncratic usage. Since this is an element of style, you are free to do whatever you wish, but remember that you may only wind up confusing your audience if you stray too far from what they perceive to be customary.

Some authors use italics to denote what a character may be thinking:

"Do you have the money?" he asked. He didn't have it. She was sure of it.

Edit: Jen is asking about how to quote something a character is writing. Here is how I have mainly seen it done, with the entire written text offset. Note that I'm not good enough at formatting in these answers to make it exact, so I'm putting everything into a blockquote and then formatting the written part as code. But on a real page the background would be all white:

Jen took a pen and began writing:

This is the first time I have written to you. I would like to ask you
if you might be interested in a manuscript that has come into my
possession, one which

She put down the pen. "Dammit," she muttered under her breath. What could she say to pique Roderigo's interest?

(The last paragraph would not be indented, but I can't seem to make that work with the formatting tools available to me here.)

The point is, representing writing on a written page is a different convention from representing speech or thought. Sometimes, if the writing consists of only a few words, a different font or use of uppercase lettering does the job:

Mary yelled, "I know what we need. A sign we can hang on the front of our clubhouse!" And she picked up a marker and a piece of cardboard and wrote: NO BOYS ALLOWED!!! She put down the marker and surveyed her work, then displayed it to the other girls. "What do you think?"