All you need to know is how to build a rank $n$ vector bundle from a principal $GL(n)$-bundle. The idea there is to view a local section of the principal bundle as a local frame of a vector bundle, and the transition functions for the principal bundle as change of frame maps for the vector bundle.

Then given any principal $G$-bundle and a representation $G \rightarrow GL(V)$, there is a naturally defined principal $GL(V)$-bundle whose transition functions are defined by composing the transition functions of the original bundle with the representation. Now use the construction above to construct the vector bundle from the principal $GL(V)$-bundle.

All you need to know is how to build a rank $n$ vector bundle from a principal $GL(n)$-bundle. The idea there is to view a local section of the principal bundle as a local frame of a vector bundle, and the transition functions for the principal bundle as change of frame maps for the vector bundle. In this case, it is easy to see that the connection on the principal bundle tell you how to "differentiate" each section of a local frame. This therefore defines a connection on the vector bundle.

Then given any principal $G$-bundle and a representation $G \rightarrow GL(V)$, there is a naturally defined principal $GL(V)$-bundle whose transition functions are defined by composing the transition functions of the original bundle with the representation. A connection on the original bundle defines a connection on the new bundle. Now use the construction above to construct the vector bundle and its connection from the principal $GL(V)$-bundle and its connection.

All you need to know is how to build a rank $n$ vector bundle from a principal $GL(n)$-bundle. The idea there is to view a local section of the principal bundle as a local frame of a vector bundle, and the transition functions for the principal bundle as change of frame maps for the vector bundle.

Then given any principal $G$-bundle and a representation $G \rightarrow GL(V)$, there is a naturally defined principal $GL(V)$-bundle whose transition functions are defined by composing the transition functions of the original bundle with the representation.

All you need to know is how to build a rank $n$ vector bundle from a principal $GL(n)$-bundle. The idea there is to view a local section of the principal bundle as a local frame of a vector bundle, and the transition functions for the principal bundle as change of frame maps for the vector bundle.

Then given any principal $G$-bundle and a representation $G \rightarrow GL(V)$, there is a naturally defined principal $GL(V)$-bundle whose transition functions are defined by composing the transition functions of the original bundle with the representation. Now use the construction above to construct the vector bundle from the principal $GL(V)$-bundle.