How to sync data between different devices
There's also Google Drive's "Application Data" folder.
https://developers.google.com/drive/android/appfolder
This has the advantage of using the user's storage space.
I would look into either Google App Engine or Amazon Web Services. They both give you free allotment of usage per month and if you go over then you start paying, chances are you wont get past the free tier for a while.
AWS is a bit more mature than GAE currently and seemed to be a bit easier to implement that GAE was when I was researching them
Take a look at the new training class for sync adapters:http://developer.android.com/training/sync-adapters/index.html for the basics of sending data from your device to a server.
On the Android device, I suggest you store your tasks in a content provider. This helps you keep track of updates since the last time you synced. You can then query the provider during your sync, send only the data that's been updated, and store the data on the server.
You should probably store the last update time on the device, so you can tell if the server contains data that isn't yet on the device. Remember that you'll have to download tasks as well if you want all devices to be in sync.