Which font should I use for the newest Unicode characters?

Since you would use just a few symbols in a special context, rather than as text characters, the practical choice is to use images.

However, if you really want to use characters, there is a very limited set of fonts to consider. According to fileformat.info, U+1F507 is supported only by Quivira, Symbola, Segoe UI Symbol, and Segoe UI Emoji. The latter two are proprietary fonts, available only in relative new versions of Windows, and as different variants (e.g., my Windows 7 lacks Segoe UI Emoji and has a variant of Segoe UI Symbol that lacks the character).

Thus, the only way that works reasonably is to use either Quivira or Symbola as a downloadable font, via @font-face. As they are rather large fonts, and you would need to serve them in different font formats for cross-browser functionality, this approach is hardly a practical option (unless you have many other special characters, possibly used in text, that also need such special fonts).

You shouldn't assume the person viewing your site has necessary fonts installed. Instead, you should add an external font. Find a font that has an appropriate licence and contains the required symbols (for example http://emojisymbols.com/), and add it to CSS as with @font-face declaration:

/*EmojiSymbols Font (c)blockworks - Kenichi Kanekohttp://emojisymbols.com/*/@font-face {    font-family: "EmojiSymbols";    src: url('EmojiSymbols-Regular.woff') format('woff');    text-decoration: none;    font-style: normal;}.controlIcon {    font-family: "EmojiSymbols";}