The definition of fiber bundle can be found from here: Definition of Fiber Bundle
Then Bredon defines Principal bundle in the exercise as follows:
I am not able to show how K acts naturally on X and the rest of the exercise.
My try was-
We need to get a map $X \times K \rightarrow X$. Let $(k,x) \in K\times X$.
Then $p(x)\in B$ and by the trivial fibration condition there exists $U \ni p(x)$ such that there is homeomorphism $\varphi :U\times F \rightarrow p^{-1}(U)$ .
Send $(k,x)$ to $\varphi(p(x),k)$.
Note: By definition of Principal Bundle F and K are same.
Am I correct?
How to prove the next part of the exercise?
I dont think you have defined an action yet. If we take $g \in K$ then we use the map $\varphi$ to define the action where if $(u,h) \in U\times K$ then $$g(u,h)=(u,gh)$$ now you need to use the fact that $K$ is the structural group to show this this is well defined.
The orbit space is $B$ since $K$ acts transitively.
If you have a section, set the values equal to the group identity element and use this to define a homeomorphism with the trivial bundle.
Both these last statement need to have the details supplied.