It depends on what you mean by proper :)
For instance: are you interested in the exceptions being thrown in your "fire and forget" calls? If not, than this is sort of fine. Though what you might need to think about is in what environment the task lives.
For instance, if this is a asp.net application and you do this inside the lifetime of a thread instantiated due to a call to a .aspx or .svc. The Task becomes a background thread of that (foreground)thread. The foreground thread might get cleaned up by the application pool before your "fire and forget" task is completed.
So also think about in which thread your tasks live.
I think this article gives you some useful information on that:
https://www.hanselman.com/blog/HowToRunBackgroundTasksInASPNET.aspx
Also note that if you do not return a value in your Tasks, a task will not return exception info. Source for that is the ref book for microsoft exam 70-483
There is probably a free version of that online somewhere ;P https://www.amazon.com/Exam-Ref-70-483-Programming-C/dp/0735676828
Maybe useful to know is that if your have an async method being called by a non-async and you wish to know its result. You can use .GetAwaiter().GetResult().
Also I think it is important to note the difference between async and multi-threading.
Async is only useful if there are operations that use other parts of a computer that is not the CPU. So things like networking or I/O operations. Using async then tells the system to go ahead and use CPU power somewhere else instead of "blocking" that thread in the CPU for just waiting for a response.
multi-threading is the allocation of operations on different threads in a CPU (for instance, creating a task which creates a background thread of the foreground thread... foreground threads being the threads that make up your application, they are primary, background threads exist linked to foreground threads. If you close the linked foreground thread, the background thread closes as well)
This allows the CPU to work on different tasks at the same time.
Combining these two makes sure the CPU does not get blocked up on just 4 threads if it is a 4 thread CPU. But can open more while it waits for async tasks that are waiting for I/O operations.
I hope this gives your the information needed to do, what ever it is you are doing :)