Exchange 2013 has better integration with SharePoint 2013, but if you're still on Exchange 2010 you can still use a tasks list in SharePoint, and view, update, add and delete tasks in that list via Outlook. One of the benefits of using a SharePoint tasks list instead of one just in Outlook is that multiple people (your team for example) can connect to SharePoint tasks list via Outlook...
If you are using Exchange 2013, ignore this next paragraph, but keep following along below it since the rest of the instructions are nearly the same.
If you are Exchange 2010 (do not do this if on Exchange 2013), you or your SharePoint farm admin need to first go to Central Admin, under System Settings > Farm Features, and disable 'Farm Level Exchange Tasks Sync'. This is enabled by default. Also, make sure 'Offline Synchronization for External Lists' is enabled - it should be already.
Then, go to the site where you have your tasks list, or where you want to create one, and go to Site Settings > Manage Site Features (under Site Actions), and if needed, deactivate 'Minimal Download Strategy', and activate 'Offline Synchronization for External Lists'.
With that done, follow these instructions to get your tasks list in SharePoint connected to Outlook so you can work with the tasks in either place, and share the tasks among an entire team, project resources or beyond.
First, on your team site, look under the gear in the top right for 'Site Contents'.
In the site contents, look to see if a Tasks list already exists and if you can use it. Or, to create a new one, under the gear again, click 'Add an app'.
On the apps page, look for 'Tasks' and click it.
Give the list a name (e.g., simply Tasks or something more unique and descriptive) and click 'Create'.
Now try adding a few tasks to your list.
From your tasks list, go up to the ribbon, click the 'List' tab, and click 'Connect to Outlook'. On the pop-ups, click 'Allow', then 'Yes.
If you have not the steps toward the top in Central Admin and Site Settings, you will still see Sync with Outlook instead of Connect to Outlook. If you are using Exchange 2013, keep following along since the rest of the instructions are nearly the same.
Once the list is connected to Outlook, you should see the tasks in your tasks view in Outlook. If it did not take you there automatically, look at the bottom left in Outlook and click Tasks.
As you can see here, my test tasks are red because they are past due.
Now you can try adding, deleting or updating some tasks from Outlook, then go back to the SharePoint site tasks list, refresh the page, and they should be updated. If you are using the tasks list for multiple people to track tasks, ensure they have rights to the list in SharePoint, then those other people can follow the same instructions to connect the list to Outlook.
You can search for specific tasks without Outlook (via the search bar on the top right of the Tasks section, or you or anyone with rights to the tasks list in SharePoint (e.g., your team), can search for specific tasks from your SharePoint search center. Do to this, go to your search center or use the search box on any page, enter some specific words or terms that might be in the tasks you want to find, then click the magnifying glass or hit Enter to run the search. When you get the search results back, narrow the results to 'Tasks' using the refinement panel on the left (you may have to click 'Show more' first to see all the result types).
The results should come back refined and tasks should be the top results. As with any search query, you may need refine your search terms to get the right results.
I hope you and your team are able to get some added productivity out of this valuable feature. Thanks for visiting.