What process do you use to select a typeface? How do you determine which types are best for long passages of text, which are best for headlines, etc.?
Readability aside, different typefaces invoke different feelings.
Before you think about what typefaces and other graphical elements to use, you have to ask yourself what the site is about and what mood you want to establish.
For example, Serifs typically are used for more traditional publications, while Sans-serif are more playful. (I'm over generalizing here).
HFJ has a great article on how to mix typefaces. In it, he categorizes each font with a feeling. I highly recommend reading it.
This is an image only flowchart. The link sends the user to a google search results page for "So You Need a Typeface." It has become quite famous so the image ought to be around for while.
Well thats a tough question to answer... You are basically saying "tell me everything there is to know about typography".
Here's a short guide. First of all you have to consider on which media your text will be read. Will it be read of a computer screen or from a piece of paper? For body text "sans" fonts (Verdana, Arial etc.) are best for on-screen reading while serifs (Georgia, Times New Roman etc.) are better on paper.
When it comes to size its virtually impossible/extemely hard to choose A perfect font size for all your users. With all the different resolutions and dpi settings it will be extremely difficult to predict how big your font will end up looking. Typically if you want to journey on the safe side you will have a font size larger than 10 pts but smaller than 14 pts. The best way to go is to make your font scalable by the user so they can decide how big they want the font to be.
Other than that you probably don't need to worry too much.