How do I get my name and picture to appear on text messages

Start at Settings > Messages > Share Name and Photo.

Recognize there are two very different types of messages. The above applies to iMessages.

About iMessage and SMS/MMS - Apple Support

That happens on the devices that you send things to. I'm not familiar with the setting, but there is a setting in Messages where you can select to share your Name and Photo. I'm not sure if that pushes it to the receiver or not. Otherwise, they have to put you in their Contacts to see your name when you send a message.