At the top box are my Toon shader nodes. Then I wanna add a normal map to these nodes. I am very new to render and nodes stuff.
On the left side of your graph, add a Glossy BSDF under the Diffuse BSDF. Mix them with a Mix Shader. Add a Fresnel and connect it to the "fac" input of the Mix Shader. Then, connect the Normal Map to all the Normal inputs you see (pictured below). Then, of course, connect the output of the Mix Shader to the input of the Shader to RGB. Change the roughness of the Glossy BSDF as needed. Note - your Normal Map should be in non-color color space (it is currently set to linear).