warning this call is not awaited, execution of the current method continues

If you really don’t need the result, you can simply change the GetNameAsync‘s signature to return void:

public static async void GetNameAsync()
{
    ...
}

Consider to see answer to a related question:
What’s the difference between returning void and returning a Task?

Update

If you need the result, you can change the GetNameAsync to return, say, Task<string>:

public static async Task<string> GetNameAsync()
{
    string firstname = await PromptForStringAsync("Enter your first name: ");
    string lastname = await PromptForStringAsync("Enter your last name: ");
    return firstname + lastname;
}

And use it as follows:

public static void DoStuff()
{
    Task<string> task = GetNameAsync();

    // Set up a continuation BEFORE MainWorkOfApplicationIDontWantBlocked
    Task anotherTask = task.ContinueWith(r => {
            Console.WriteLine(r.Result);
        });

    MainWorkOfApplicationIDontWantBlocked();

    // OR wait for the result AFTER
    string result = task.Result;
}