Our very favorite project management software. Everything you would want or expect in task management. Also, it is cloud-based, has mobile apps, keyboard shortcuts, now offers Kanban and Gantt views if that's your thing. Highly integrable. Too many features to even list, and they're constantly shipping new ones, all without overloading the UI and making the tool clunky.
Not a good place for highly structured customer data. Keep that in your CRM and link to it from Asana. Asana is also so easy to use that you tend to see lots of disorganization and mess, because you don't have to be so deliberate in setting things up.
Sync your data to Salesforce and track your team as they move work forward. Set goals for timelines, response/cycle times, work in progress, and more. Any process, any object in Salesforce.