We use Harvest
to track our time and it's got some pretty good tools for planning, reporting and invoicing as well as some useful API integrations (we sync with Trello). We use it primarily because our work is billed to clients so I'm a little wary of your comment:
'in order to ensure we are efficiently working through our assigned tasks'
Tracking time doesn't and never will ensure you do anything other than record time (and use up some of your team members precious time in doing so). It will also never tell you how effective you are.
If you're interested in understanding your teams efficiency then assuming you break down work into work items (such as user stories) then the simplest thing to do is record how many things your team gets complete (done done, deployed, whatever) each week and measure that week on week. I'm not about to explain the details here - there's a plethora of Agile resources and there a blog post I wrote here
To reiterate: we use time tracking software to track time so we can bill for our work, I wouldn't (and couldn't) advocate using time tracking software to understand efficiency or effectiveness.