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I currently have 4 card (1 discover secured, 1 Amex, 1 Cap One, 1 Chase Freedom Authorized)
I want to get a CFU first because getting one more will limit me from 5/24 rule.
I also want to get Amazon Chase Prime card because of how much point I can earn.
How long should I wait between each application? or should I apply only one of those card?
If your profile is strong enough, you may actually want to apply for the Sapphire Preferred instead of the Freedom Unlimited. The CSP has no annual fee in year 1 and has a higher sign up bonus. If you decide against keeping it in year 2, you can downgrade to the FU.
That said, you can actually apply for two cards on the same day. I applied for the CSP online and then a few minutes later applied for the Amazon Prime Visa online. I was approved for the CSP but denied for the Amazon due to 'too many recent requests for credit/new accounts' - including other banks and a business card, it would have been my 6th new card between July and January. That's what seemed to get me the denial for the Amazon. By applying for both on the same day like that (online at least) you will combine hard pulls so the second app is something of a 'freebie'.