When I zoom in to color in Photoshop it gets so pixelated like in the picture below. Also when I want to draw or trace on a picture, It doesn't draw between those squares and just shift the line to the nearest square corner causing bad quality! Is it normal due to my screen resolution? is there any way to improve this? Thanks a lot.
Photoshop(PS) is a bitmap editor, it edits pixels. There is no info between the pixels. This is true even to vector content authored in Photoshop. So when you zoom in Beyond the documents native pixel dimension that is what happens.
Note this is not nesseserily the way it outputs on to say a print medium as the printer probably does not use nearest neigbourhood interpolation (PS does so you know what is in the sample).
Is there a way to improve the situation?
Well, it depends on what you mean by this. Yes you can make a bigger document. But in general is a bit hard to say as you do not describe what your purpose is. If you are preparing for a screen with this resolution itseems a bit counterproductive. Or maybe you want to work in Illustrator?
If you zoom to 800% or more, PS will show a pixel grid. This is a thing they added maybe in CS 5.
You can turn it off from the top menu:
View > Show pixel grid