Looking for something like Trello

I’m looking for an app like Trello that will allow integration with calendars and reminders.

Sounds like a job for Curio.
Curio supports metadata on everything, so an item can have a due date, reminder, etc.
I use it to track participants for my studies, from recruitment to final payment.
Here is a post on the Zengobi forum about this workflow.

Trello already does this with Power-ups and/or integration with Zapier.

If you wait a few weeks (?) Todoist, which already has two-way integration with Google Calendar, is introducing a Boards view.

This is something I’m really looking forward to, as I use Todoist and Trello and Google Calendar (but you can sync GC events into Apple Calendar fairly easily too).