Search engines won't ignore or penalise quotations, provided they're accompanied by original content (i.e. the content is not wholly or substantially a duplicate of pre-existing web content).
In fact, proper use of
<blockquote> elements (with
<cite> where appropriate) is arguably a good way of indicating to search engines that you're being honest about material you've taken from elsewhere.
There's no SEO difference between
<blockquote>, but they do have slightly different purposes. I'd encourage you to consult the latest version of the HTML specification.
A pull quote is a visual style and not syntactically distinct from
<blockquote> in terms of document structure, so there's no special HTML markup for those.