Consider a ringed space $(X,\mathcal{O}_X)$ and a coherent $\mathcal{O}_X$-module $\mathcal{F}$ which is generated by global sections. Here we use the definitions in Stack Project 01AL and Stack Project 01CF.
It is easy to see that the tensor product $\mathcal{F}\otimes\cdots\otimes\mathcal{F}$ is generated by global sections.
Are the exterior power $\wedge^n\mathcal{F}$ and the symmetric power $S^n\mathcal{F}$ also generated by global sections?
Yes. Globally generated means you have a surjection $G=O^k\to F$ for some $k$. Then $\wedge^n G\to \wedge^n F$ is onto. But $\wedge^n G$ is just a direct sum of $O$s and thus $\wedge^n F$ is globally generated. Similarly for the symmetric power.