How to make permanent design on my plain white T-shirt?
If you are happy to hand draw, you can get permanent markers specially made for fabrics, which will stay in even if washed often. These also come in many colours and in several surface structures.

Less fancy but often nearer at hand, normal permanent markers, like sharpies. These will fade a bit with washing but mostly good for several to many wash cycles.

If you do not want to hand draw, the easiest option is to print or have printed on iron on sheets. This website has instructions, which include one brand of sheets. (I am not connected to the website and have never used this method and as such do not know which sheets and which printers would be best to use.)
The commercial alternative is a T-printing service, often they start with very small numbers but the price goes down when you get more printed.  

More permanent, harder to do yourself and more expensive to have it done:
Embroidery.
Mostly machine embroidery which is done from computer pictures. But you can hand do it with enough patience (and a lot of trial and error.)
You can do hand embroidery as well and I am sure that this stack exchange site has questions about it.