Make sure you install the font.

To change your font to Fira Code, open Atom's preferences (cmd+, on Mac or ctrl+, on PC), make sure the "Settings" tab is selected, or the "Editor" in Atom 1.10 and forth, and scroll down to "Editor Settings". In the "Font Family" field, enter Fira Code.

If you wish to specify a font weight, specify the weight following a space (if Windows, e.g. Fira Code Light) or as FiraCode-Light (if Mac).

Ligatures are enabled by default in Atom 1.9 and above.

To enable ligatures on older Atom versions (Atom 1.1 and newer), go to Menu → Stylesheet...

and add text-rendering to atom-text-editor:

atom-text-editor {
  text-rendering: optimizeLegibility;
}

Beware that in some syntaxes selected ligatures might not work. This is usually a syntax parser/tokenization issue (e.g. -> breaks into two symbols by JS/Ruby syntax highlighter). See issues #63 and #69

To turn off ligatures inside of strings and regular expressions you can add this to your stylesheet:

atom-text-editor.editor .syntax--string.syntax--quoted,
atom-text-editor.editor .syntax--string.syntax--regexp {
  -webkit-font-feature-settings: "liga" off, "calt" off;
}

If there is any other place that you find you do not want to see ligatures, place your cursor on the location, hit Ctrl+Alt+Shift+P and add the bottom-most selector listed in the popup notification to the rule above.