Would you put the name of the Mayflower in quotations?
Yes, you would put "Mayflower" in quotations because it is an actual name of a boat.
No, you would italicize or underline the name of a painting. If you are using MLA format, then you should underline the name. The way I remember is that quotations go around short works like poems or plays, whereas sculptures and paintings take a long time to create. (Debatable, yes, but it's a helpful mnemonic.)
You put quotations in when someone is talking, or when you are state something that is not really something. Here is an example: "That is so hard! How do you do that?!" Screamed Sally. Another example: "You better get your chores done, Michael." Said Michael's dad unhappily. Like an example would be: I jumped in my "bed". If you call a bed a blanket and a pillow.