iPads won't send SMS messages by themselves. They need to be paired with your iPhone. You probably did this with the old iPad so long ago that you don't remember.
On your iPhone in Settings > Messages select Text Message Forwarding and make sure your new iPad is turned on.
On the iPad in Settings > Messages under Send Receive make sure that Start New Conversations From is set to your phone number. This is not necessary to use SMS on the iPad, but it does help keep conversations from getting fragmented.
Background: Messages on the iPhone and iPad's use two different message services. iMessage is Apple's service. It has a lot of extra features but only works with an internet connection, and only with other Apple devices. SMS (and MMS) is the standard cell phone message system. It does not use the internet, but it works on all phones. iPads and other non-phone devices need a paired iPhone to send and receive SMS messages in the Messages app. They all need to use the same Apple ID/iCloud account.
Apple devices will use the more capable iMessage system when sending to other Apple devices. They fall back on SMS when sending to non-Apple devices.