Which one is a better approach Sorry for this beginner like question, but I can't find the answer since I don't know what I need to type in Google search. I have been learning to use Blender for the past few months, and I tried to make some low poly wooden fence for small houses

At first I created the standing part (I don't know what i should called it, sorry for my bad English)

and then put a plank right through the middle part

While it looks good, there are some parts of the plank that driven into the standing part, while it's not visible, I believe it's still rendered by the engine.

Since it's only a small object and few in number, I believe in won't effect the performance. However, in the future, I was hoping to develop my own simple and low poly game. If I ever used this as an asset for my game, will it affect the performance altogether?

After some thoughts, I separated the plank into parts, and placed them between the standing parts. Since it's a simple type (cube shaped) fence, it doesn't take much time to connect the vertexes together, but what if the fence is round type? I have tried to make a hole in the standing parts, connecting the vertexes, and it's time consuming. 

My question is: considering the amount of time needed to work on, and how much it's affecting the render time process, would you rather drive a cylinder type mesh into the middle part, or doing the same thing as I did for the cube shaped fence?
When it comes to game assets there is one general simple rule. Use as many polygons as necessary and as little polygons as possible. No exception.

It's not only the amount of polygons but also the amount of vertices that matter. On top of that you also have to take the amount of UV islands for texturing (that means no "Smart UV Projection" or other fancy UV shortcut nonsense). 

Also, don't assume the performance hit by the size of the object (in terms of screen space) in your finished game, neither the CPU nor the GPU care about that. They only care about the complexity as I described it above. 

Continuous planks and bars (like in your fist example) through other meshes are the way to go, as long as they stay at the same position at their entry and exit points.