No, you don't have to pay APR if you pay on time and in full every month. And your card most likely has a grace period. A grace period is the length of time after the end of your billing cycle where you can pay off your balance and avoid interest. To take advantage of a grace period, you need to pay for all your charges every single billing cycle. If you don't do that one month, you'll lose your grace period, and your charges will start accruing interest right away. You'll have to pay in full for two consecutive billing cycles to get it back.
So paying on time won't get you out of paying interest on its own. You'll just avoid paying late fees and hurting your credit score. You have to pay in full if you don't want to pay interest.
Here's how to avoid paying APR:
If you pay your bill in full by the due date every month, you won't pay any interest, thanks to the grace period most credit cards have.
A credit card's grace period typically is the time between the end of the billing cycle and the due date.
If you lose your grace period by carrying a balance past your due date, you can get it back by paying your bill in full two months in a row.
Balance transfers and cash advances don't have grace periods, so interest will begin piling up as soon as you make the transactions. The exception is if your balance transfer has a 0% introductory rate for a certain number of months.
Getting a 0% Intro APR credit card gives you a larger window of opportunity to pay off your credit card. It allows you to pay off your balance before the end of the promotional APR period without accruing interest. You'll still need to pay at least the minimum payment on time and any remaining balance that hasn't been paid off by the end of the introductory 0% APR period will accrue interest at the card's regular rate. It's also a good idea to pay more than your minimum payment every month, as leaving a big balance on a credit card for a long time can cause a dip in your credit score.
The simplest way to handle things is to set up automatic monthly payments for your full statement balance from a bank account. Just make sure that account's balance doesn't get too low.