The easiest way to start a Chat with someone is from their Profile page. On the top-right side, right above their latest posts, you'll find a button with a Text Bubble icon. If you click it, you'll be redirected to the Chat page.
On desktop you can also hover over someone's Username and you will find a Chat button in the tooltip that shows up.
Keep in mind that you also have a Chat button present on Desktop at all times, at the top of the page. This will simply open to the latest Chat you've had, so don't get it confused with the one from the Profile you're viewing.
You can see who your Chat partner is currently at the top of the page, as well as a list of Partners on the left side.