Will my customers be taken to another website to pay online?

It depends on the gateway:

• Razorpay opens its payment window on top of your checkout, so the customer stays on your site.
• Stripe, PayPal, PayU and Cashfree send the customer to the gateway's secure page and bring them back to your order confirmation page after paying.

Checkout shows "You'll be redirected to complete the payment securely." when an online method is selected, so customers know what to expect.

If they close the gateway page without paying, the order is kept as Pending and their cart is not emptied, so they can try again.