Here at Miniwebs, we use "Asana" - https://asana.com
Each of our customers have their own 'Project' and whenever a website update task comes in from a client, a 'Task' is created under the Project.
That task can have a due date assigned to it and all tasks can be sorted by due date, or Project, however we wish.
But not only that - you can have other staff members on your team and assign projects and tasks to them as required.
Recurring tasks can be allocated - they just pop up onto todays task as they become due.
But like everything, it's all trial and error but I highly recommend Asana to any business.
What's more - it's FREE! There are a few limitations of course - eg number of team and templates.
At Miniwebs we have only just moved to the Premium version after 5 years of using the FREE version.
Give it a go! Once you get the hang of it - you'll love it.