Yes and yes.
And to text people from your laptop you use the Messages app. In the Messages app you sign in with your Apple ID. So, you would sign into Messages on your laptop, then sign into Messages on your new iPhone. When you sign into Messages with your Apple ID, it activates Messages. On the iPhone, the phone number of that phone is always selected, and you can add additional email addresses if you like under the Send & Receive option.
Once you have done that, it automatically adds those phone numbers and emails to your Mac, and you select which ones you want to use under Preferences in Messages on your Mac.
You also have an option in the Messages app on both the iPhone and the Mac to choose which of the phone numbers and email addresses associated with your Messages log in you want to use to "Start new conversations from", so you can make them both the same via that option.
I've attached the Messages support links for both the Mac and the iPhone so you can read up on how it works together:
https://www.apple.com/support/mac-apps/messages/
Use Messages with your iPhone, iPad, or iPod touch - Apple Support
Best of luck,
GB