Did you enable iMessage on the new phone?

Go to Settings/Messages and make sure you've enabled iMessage. Use your Apple ID to enable it if you haven't, that's the fastest way to get it working on your phone. You have to re-enable it on new devices, sometimes when you change SIM cards, etc. Test it by trying to send to somebody you know that has it working already. If it's not working, then try disabling and re-enabling it, using your Apple ID. You can even try sending one to yourself.

Messages is an app, it supports SMS and iMessage protocol. You can view it as IOS' "unified" messaging app. It will use iMessage if it's enabled and the recipient supports it, and will be indicated by greyed out "iMessage" in the empty text box and blue text. Otherwise it will show "text message" greyed out in the empty text box and use green text.

iMessage is a messaging protocol, supported by Apple and uses your wifi or data connection.

You will also may want to edit the email addresses and numbers you send/receive iMessages from in the same Settings, you can also choose which number or email address new iMessages are sent from.

-David