Chern class of $E \times F \to M \times N$ Let $M$ and $N$ be two complex manifolds and $E \to M$, $F \to N$ be two holomorphic vector bundles. 
1) Is there a way to define vector bundles $G \to M \times N$ such that a fiber over $(x,y) \in M \times N$ is, eg, $E_x \oplus F_y$ or $E_x \otimes F_y$ ?
2) How are the Chern classes of such bundles related to the Chern classes of $E$ and $F$?  
 A: Consider the projection $p_M:M\times N\rightarrow M$ and $E_M$ the pullback of $E$ by $p_M$. The fibre of Withney sum of $E_M$ and $F_N$ is $E_x\oplus F_y$.
Now, use axioms 2 and 3 here to compute the Chern classes of $E_M\oplus F_N$ with the Chern classes of $E$ and $F$.
https://en.wikipedia.org/wiki/Chern_class#Classical_axiomatic_definition
A: If $p_1:M\times N\to M$ and $p_2:M\times N\to N$ are the projections, then you can pull back to get bundles $p_1^*E\to M\times N$ and $p_2^*F\to M\times N$. Then we can define the "box product" $E\boxtimes F\to M\times N$ by
$$E\boxtimes F=p_1^*E\otimes p_2^*F.$$
Now, if $(x,y)\in M\times N$ then since $(p_1^*E)_{(x,y)}=E_x$ and similarly $(p_2^*F)_{(x,y)}=F_y$, we see that the fiber of $E\boxtimes F$ over $(x,y)$ is equal to $E_x\otimes F_y$.
You can do the same thing for the direct sum, as someone else has already suggested; I don't know if $E\boxplus F$ is standard notation for this, but it makes sense to me.
How does relate to Chern classes? Well, by definition $p_1^*(c(E))=c(p_1^*E)$, similarly for $F$, and for the sum you can calculate
$$c(E\boxplus F)=c(p_1^*E\oplus p_2^*F)=c(p_1^*E)\smile c(p_2^*F)=p_1^*c(E)\smile p_2^*c(F)=c(E)\times c(F)$$
where the latter is the "cross product" or the "external cup product". Finding a similar formula for $E\boxtimes F$ will be more difficult, as I don't think there's a nice formula for $c(E\otimes F)$ in general unless the two are both line bundles.
