I have a Lie group $G$ acting on a manifold $M$ in a continuous way (respectively smoothly). $$\Phi:G\times M\rightarrow M:(g,p)\mapsto g\cdot p$$
Why is the following continuous (respectively smooth) $$\phi_g:M\rightarrow M:p\mapsto g\cdot p$$
Why is the following smooth if the action is smooth $$\phi^{(p)}:G\rightarrow M:g\mapsto g\cdot p$$
PS: It should be fairly obvious. Is there such a property as restriction of a continuous/smooth function is continuous/smooth?
The fact that you appear to be missing is that the inclusion map $X \to X \times Y$, $x \mapsto (x,y_0)$ between arbitrary spaces (resp. smooth manifolds) for any fixed $y_0 \in Y$ is continuous (resp. smooth). Then your maps are simply compositions of inclusions with the continuous (resp. smooth) action map.