First off, do not copy and paste from Word into the CSS editor window. Word adds all sorts of garbage code and changes key characters used in CSS into other things (single and double quotes to name two) that will mess things up.
Always copy and paste CSS from a plain text document from Notepad or TextEdit or something like that.
The main font for the theme is set in the body section of the CSS, but then the font for different page elements are set in the controlling selelctors for that element (if the font is going to be different from that set in body), so yes you will have to change the fonts in other locations as desired.
Two notes of caution though:
1. Do NOT paste the entire stylesheet into the CSS editor. That will more likely than not mess up your site. Paste only the specific selectors, and the specific declarations you are changing.
2. Do NOT use oddball fonts that you have on your computer. Browsers pull fonts from the visitors computer, and if they don't have that font, then the browser will substitute either "serif" or "sans serif and your site will not look like what you intended.
You can use typekit fonts and then with that service, the fonts you use will be served to the visitor's browser by typekit, so they do not have to have the font installed on their computer.