I paid my credit card on due date. Then later I purchased something on the same day. I'm curious what will happen? Do I need to pay that i purchased on the same day?
No, you do not need to pay it on the same day. That new charge will appear on the next monthly statement you receive. Your billing cycle is offset so that the cycle ends and a due date is provided. What is owed on your due date is charges during the previous billing cycle, not charges up to that exact date.
If your billing cycle is Oct 1 - Oct 31, all charges accrued during that time period will be on that bill. The due date for that bill might be Nov 16. The next cycle is Nov 1 - Nov 30, so charges made on those dates would be due Dec 16.
These dates are arbitrary and just provide an explanation. Obviously you need to double check your due date and cycle dates for your own card.
No, generally the credit card company present you with a monthly bill or statement (paper or online) and you pay the amount on that document.
Anything you purchased recently will be on the next bill, due in a month's time, and typically will not attract any interest charges before then.
Typically if you withdraw cash, it attracts interest immediately, but you are not expected or required to pay it off until it appears on a subsequent statement/bill.