Restore your new phone from a backup of the old phone. That way you should have all your chat contents back.
If you changed SIM cards and have a new number, start WhatsApp, go to Settings / Account / Change number and setup your new mumber. Your contacts will get a message in their chats with you that you have changed your number.
If you did not change your number, you can start using WhatsApp as before. Your contacts will see a message in their chats with you that your encryption key has changed.