The web and desktop apps have an emoji button, right next to the attachment button, within any conversation. You can also access emojis through a search/auto-complete.
To search/autocomplete emojis, first type a colon (:). You will see a dialog box appear above where you are typing. Then, just start typing a description of the emoji you want ("happy" for example). You can click the results in the dialog or use the arrow/enter keys to select the emoji you want to insert. This is similar to how Github and Slack manage emojis, if you are familiar with them.
Some people will just want to use the normal smiley faces, rather than the emoji results. Since autocompleting the emojis begins with a colon, this can get in the way. You can still use your legacy smileys. To dismiss the emoji search, just hit the esc button or add a space after your smiley face.
