The out of the box view designed for this is the Resource Assignment view. It is a ribbon option on the Resources page. Select a resource(s) and then select Resource Assignment. You can also create views with filters or create a resource view by manager to simplify the process. Hope that helps.
The simplest way might be to use a Task Usage View. If the resources have Departments assigned you can group by the department values to consolidate related resources together. I would filter the view to remove any completed tasks. I would click on the Start column heading and sort the start dates in ascending sequence. An finally, I’d set the time phased data (right side of the view) timescale to show weeks. You can also shift the time phased grid so the current week appears at the leftmost side of the time phased view.
This will group all resources for each department, remove any “noise” caused by completed tasks, and then all remaining work on started tasks plus any new tasks starting in the current week would appear at the top of the task list under each resource.
I’m sure there are other ways, but this is where I’d start.