Keep using your American Express Costco card through June 19th (Sunday)
Start using your new Citi card on Monday, June 20th.
Your Costco rewards will be carried over to your Citi card automatically (according to Costco).
If you received your new Citi card, activate/check-in your card. You can do so by calling or getting online. It took me about 5 minutes (or less) to check in online.
I don't know about you, but I have absolutely no reason to use American Express card anymore. I've alway thought American Express was hoity toity. Costco was the only reason I opened the account.
PS. Oh, Christmas is coming to costco in 7 weeks. :)