In the Timely iOS app you can easily see what messages from your clients have not been read. There is a count of your unread messages on the menu bar, and unread messages are indicated with a red dot in the message list.
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.
If you tap Edit you can select multiple messages to bulk mark messages as read or unread.
You can also mark messages as read or unread from your SMS replies list. Again, just tap Edit and select the the messages that you want to mark as either read or unread.