You have two options:
(1) Use Messages in iCloud, which ensures that all devices that are connected to your Apple ID will display the same Messages. They will be synchronized.
(2) Use Message forwarding on the iPhone to all the other devices. Here is a little checklist for setting up (2) with an iPhone and a Mac. The process is similar for other devices in addition to the Mac:
iMessage checklist
iPhone and Mac both logged in to same Apple ID. Try logging off and logging on if necessary.
iPhone Settings => Messages shows iMessage as "on" (sometimes toggling this off and then on fixes some things)
iPhone Settings => Messages shows Send & Receive showing both phone number and email (Apple ID)
iPhone Settings => Messages shows Text Message Forwarding and your Mac is listed and shown as "on" there.
iPhone Settings => Messages shows Send as SMS "on" and MMS Messaging and Group Messaging both "on"
On the Mac: Messages => Preferences shows same and correct Apple ID and "You can be reached for messages at" indicates both phone number and Apple ID (email), and "start new conversations from" indicates the phone number.
* Don't see the Text Message Forwarding setting on your iPhone? Go to Settings > Messages, turn off iMessage, turn it back on, tap Send & Receive, tap Use Your Apple ID for iMessage, then sign in with the same Apple ID used on your other devices.
How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac - Apple Support