How do I know when my text messages Been Delivered and read

To see if it's been deliverer, touch the message and drag from right to left, which shows the time it was sent from your phone.

For read notification:

1. Open Settings.
2. Tap Messages (the green icon with a white text bubble inside it).
3. Turn on Send Read Receipts.
4. Others are notified when you read their messages. If your recipient also enabled read receipts, you'll see Read beneath your message along with the time it was read.

Hello,

For iMessages, go to Settings > Messages and turn on Send Read Receipts

Also see:

About iMessage and SMS/MMS - Apple Support

and this:

Use Messages on your iPhone, iPad, or iPod touch - Apple Support

Good luck.