How do I change the "Posted By" line of each post?

Note: This article only applies to blogs using a classic template. If you're using Layouts, you can simply click the "Edit" link on your Blog Posts page element.
If you wish to change your Post footer in one of Blogger's default templates, go to Template | Main Template and scroll until you see this code:

<div class="byline">

within the <Blogger> and </Blogger> tags. Anything between it and the closing </div> tag is the footer.
Let's dissect it piece by piece. Here's the default byline, in published form:
The first part (the "#") is called the permalink; this is code that generates it:
The text inside the " " (quotation marks) generates the link, <$BlogItemPermalinkURL$> being the link to that specific post on the archive page. The # sign at the end, just before the </a>, is the link text (feel free to change it to whatever you wish, including another template tag, <$BlogItemDateTime$> for example).
The second part, "posted by etc." is generated by this text, and can be modified as you wish:
Notes:

• A permalink can also be displayed in the following manner:

  <a href="<$BlogItemArchiveFileName$>#<$BlogItemNumber$>" title="permanent link">#</a>