I have a locally trivial bundle $B$ with base $X$ and projection $p$. I have open cover $\{ V_{j}\}$ and "charts" $\phi_{j}: V_{j}\times F \to p^{-1}(V_{j})$.
(*) On each $p^{-1}(V_{j})$ I can get topology induced by $\phi_{j}$ and on $p^{-1}(V_{j}) \cap p^{-1}(V_{i})$ the maps $\phi_{i},\phi_{j}$ induce the same topology.
What exactly is topology on $B$ induced by $\phi_{j}'s$?
Is it the one that $U\subseteq B$ is open iff $\phi_{i}^{-1}(p^{-1}(V_{i})\cap U)$ is open $\forall i \in I$, or is there some other natural topology?
The topology on the total space $B$ can be thought locally as a product space of $F$ and an open set in $X$.
As you guessed, $$ \begin{align*} U \text{ is open $B$} &\iff p^{-1}(V_i)\cap U \text{ is open in $B$ for all $i\in I$} \quad \text{($\because$ $U=\bigcup_{i\in I}\bigl[p^{-1}(V_i)\cap U\bigr]$) } \\ &\iff \phi_i^{-1}(p^{-1}(V_i)\cap U) \text{ is open in $V_i\times F$ for all $i\in I$} \quad \text{($\because$ $\phi_i$ is a homeomorphism) } \end{align*} $$ but the condition is not convenient to use.
Actually it may be better to start with two topologies given on the total space $B$ and the base space $X$, and a continuous surjective projection $p\colon B\to X$.