Hi Nathan,
Thanks for getting in touch. You would need WooTickets if you wanted to do any kind of event registration or ticket sales. However, you do not need Events Calendar PRO unless you want the PRO features like special widgets, recurring events, extra views, and support. Otherwise you can use WooTickets with the core free plugin, The Events Calendar.
When they select the date they can choose one of the predefined times and purchase a ticket.
One option for something like this would be to use Events Calendar PRO, which includes a Day View. Your users could click to see events for one day, divided by time. You could also do it with just The Events Calendar by listing the times on the Calendar View. However, please note that there is no feature that allows a user to choose an event on a specific day and then choose times from that page. Different times need different events on the backend, even if the event itself is identical. Does that makes sense?
Cheers,
Leah