Moved from PHP; thread title edited to be more specific.
First step is to remove the <blockquote>s. You can't just remove everything between the opening and closing tags because you could potentially remove stuff you wanted to keep.
Recursion to the rescue. Basically,
1. Find a <blockquote>
2. Look for a bunch of stuff
2a. It can have something that isn't an HTML tag
2b. It can have something that isn't an opening or closing <blockquote>
2c. Or it can try to match the entire regex again right there
3. Find a closing </blockquote>
That will remove quotes so long as they're paired. If the opening and closing tags don't match up then you'll be stuck with them...
Which is one reason you next grab
the stuff between the <p>s, not just try to remove the remaining HTML.