Hello Link3,
It sounds as though you're wanting to sync your messages with iCloud, and use text message forwarding. From the message that you're receiving have you verified that your iPhone is currently connected to a charger and connected to Wi-Fi? If not, you will want to have your iPhone charging and connected to Wi-Fi as that is required for messages to sync to iCloud. If your iPhone is disconnected from either the sync would pause.
If you continue to have issues syncing your messages you will want to confirm the Messages in iCloud settings as outlined in the following article.
Use Messages in iCloud
For Text Message forwarding you'll want to make sure that you're signed in to iMessage on both your iPhone and Mac, and that your mobile number is added to iMessage. You can find the steps for setting up iMessage in the following articles, along with adding your phone number, and setting up Text Message forwarding.
Set up Messages on iPhone
Use Messages with your Mac
Add or remove your phone number in Messages or FaceTime
How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac
Have a great week!