Vertical lifts on vector bundles

Question

So let us assume we have a vector bundle $\pi:E\rightarrow M$ and $\nabla$ a connection on $g$.I know that if $v\in E_p$, then there is the standard embedding of $v$ in $T_eE$ where $\pi(e)=p$. Namely we can send $v\mapsto [e+tv]$. While this definition works, it is not practical for computation. So I was wondering if their was an alternative definition for the vertical lifts. In particular it would be great if someone could discuss the construction of the horizontal and vertical lifts. I am still a bit confused about the construction. In addition I am unsure why given a vector field $X\in\Gamma TM$ its horizontal lift $\tilde X\in\Gamma TE$. I've been looking at Peterson's text and it seems he omits some details.