Make sure your new account is open.
Login to your old bank account online. Review your transactions. You're looking for:
Transfer your payments and deposits to your new account. You'll need to call your providers and employers directly or contact them online.
E-transfer your money from the old account to your new account.
Go to your old bank in person - some banks charge fees if you close accounts online / by phone.
The banker may ask you why you're closing the account. Tell the truth – that you found an offer that suits you better. The banker may try to match the offer - it's up to you to decide.