How can I see which language type will result from the union or intersection of different language types?
This notion is clalled closure property. A family $\mathcal F$ of languages is closed under (binary) operation $\#$ if $K\# L$ is in $\mathcal F$ for all $K,L$ in $\mathcal F$. 

There are no general rules: context-free $\mathsf{CF}$ is closed under union $\cup$, deterministic context-free $\mathsf{DCF}$ is not. However there are resources that list the known closure properties for important families, especially the Chomsy hierarchy.