Emmie, In my experience, which covers quite a few banks in quite a few countries, you will probably have to validate the new card reasonably quickly after receiving it. (This is usually just a simple matter of using the new card to buy something (in a shop, not online) or withdrawing cash at an ATM.)
Don't update the card details anywhere until you've done this, as some companies will check the new details online when you inform them. If they do this before you have validated the new card, they will receive a refusal, which can pose problems all round.
You will normally need to update online accounts where the supplier, e.g. Amazon, charges your debit (or credit) card directly, even if the 16-digit card number hasn't changed. (The magic number on the back (the CVV) will have changed as this depends on the expiry date, amongst other things.) You will not need to update people who hold Direct Debits as these are charged directly to your bank account, not via your card. You can normally see which is which on your bank statement: mine shows DD for Direct Debits & DEB for a debit card payment.
As for which of the two cards to use, it's quite normal for the old card to be refused once the new card is validated, so I cut my old one up once I have validated the new one.