Inside, the user profile, I am trying to change my time zone to something else (say, UTC + 1:00)
Then I placed a label inside a test module and wrote DateTime.Now.ToString(), but it is not showing me the time according to the time zone that I set.
Am I doing something wrong?

Even if I change the timezone from Admin > Site Settings > Advanced Setting. Then also it is not showing me the time according to set time zone. IS my understanding incorrect? IS that just for just save as the "setting"?

I think that DateTime.Now.ToString() returns the current SERVER time. That is an ASP.NET method, not a DNN method. You'll need to dig through the DNN API to find the correct method to use. A good way to do this would be to look at something that displays the user's time, and then look in the code to see what is being called.

There is a compiled help file for the DNN API on the page where you download DNN.