charging the interest rate? I have Chase and Capital One

Usually, if you don't pay the entire balance of the card on the due date of the bill, you will get finance charged on the balance you are carrying over to the next billing cycle.

You have one month. If you haven't paid the balance due in one month, they start charging the interest on the remaining balance.

Unless you have what is called a "grace period" on new purchases, you are charged interest from day one. The term of a grace period - if you have one - varies from account to account, depending on your credit card company. Some agreements give you until the next billing date to pay the balance, some till the end of the month, some even 30 days.
You'll have to check your terms and conditions for the exact time for your card.

Immediately.

Typically you get a bill showing total of what got added based on your purchases in the last month, and there is a deadline to make a payment. If you do not pay in full by that deadline, then the next month bill will show the previous balance plus interest charges on that prior balance.

They had wanted you to make a partial payment on the balance. If you had failed to do so, then in addition to interest, they charge you penalties.