You can check your message status on the conversation page.
When you send messages to people using TextMe, there are 3 different statuses at the bottom of your message;
"Sent" means that your message was successfully sent by TextMe.
"Delivered" or "Received" means that your friend receives a message notification.
"Read" means that your friend opens your message.
When you send SMS to people not using TextMe, you just get the notification that message is sent.