In the Timely iOS app you can easily see the status of your client messages, such as whether they have been read or not. There is also a count of your unread messages in your menu bar, with unread messages indicated with a red dot.
If you tap on a message, you will see a full message history with your customer. You can tap on any message from your customer to either mark it as read or unread, you can also select multiple messages in order to take bulk action.