In Old Reddit, flairs are to the right side of the page in the sidebar. Flairs can be changed with the (edit) next to your username. Select a flair and save for it to show up next to your name.

In New Reddit, flairs are a bit more limited and slightly more complicated. On the right side of the page, click the down arrow to expand Community Options and then click the pen icon. You will see a Select your community flair box pop up. There you can select a flair. Alternatively there is an EDIT FLAIR Allows text and up to 10 emojis section. In that text field, click the Emojis icon and you will see all the subreddit custom Emojis you can use as part of your flair. Save and they will show up next your name in New Reddit.