Is pilgrims a proper noun? What about New World?
Pilgrims is not a proper noun. New World is.
Pilgrims is a plural (talking about more than one) common noun because it is general (not specifying one particular pilgrim and not capitalized unless it is starting a sentence).
The New World is a proper noun because it is referring to a place by the place's name (at the time, since we no longer call it the New World). Examples of proper nouns are names of people (Benjamin Franklin, Andrew Jackson, etc.), places (Tokyo, San Francisco, Marseilles, Staten Island, Asia, Liechtenstein, etc.), and monuments (Eiffel's Tower, Statue of Liberty, etc.).