Is it possible to redirect to a new page and display booking details after submitting a booking?

You can enable the "Redirect to thank you page" option on the Booking > Settings General page in the "Booking Confirmation" section. This allows visitors to be redirected to a specified page after creating a new booking. To showcase the confirmation and "booking details" on this page, simply insert the following shortcode:

[booking_confirm]

For guidance on inserting Booking Calendar shortcodes into a page, refer to the instructions on this page, particularly in the 4. Manual Shortcode Configuration section.