How can I get a specific Device ID, or Task ID?

Open the target Device in the list to Edit. In the browser’s address bar you’ll see something like:

https://user.dotcom-monitor.com/ClientID/DeviceEdit?pid=dc7f4ff2ca944dekjh1078b96707002& deviceId=63698 & taskId=132834 

The deviceId=63698 is the device ID.
The taskId=132834 is the task ID.