As others have pointed out, there are multiple ways to handle this in EE. And, the Partials approach mentioned by @dadonbike is generally the best, modern, architectural approach. But this may require you to make a bigger mind-shift coming from Wordpress. And it also can't be done without add-ons / it's not an approach that's native to EE (which actually is a comment on EE, rather than this approach--but it's worth mentioning).
A simpler approach is to use conditionals to display different chunks of the template, depending on the context as determined by the URL segments. And, a variation on this is to keep each context's template separate, but embed (include) each template conditionally into a master index file. Here's an example (that's a blog/index template):
{if segment_2 == ""}
{!-- MAIN PAGE --}
{embed="blog/.mainblog"}
{if:elseif segment_2 == "category" AND segment_3 != ""}
{!-- CATEGORY PAGE --}
{embed="blog/.category"}
{if:elseif segment_3 !=""}
{!-- ARCHIVE PAGE --}
{embed="blog/.archive"}
{if:else}
{!-- PERMALINK PAGE --}
{embed="blog/.permalink"}
{/if}
Because these are URL-driven approaches, in your case, if News and Events really warrant their own URLs structures, then that's a natural reason to not combine them into a single set of conditions.
Generally, if your conditionals start getting unwieldily, you'll wish you'd split things into separate templates. But, the flip-side of more separation is often duplication, where you have very similar or the same code copied in multiple templates.
So, the Partials approach does a great job of cleaning-up / removing conditionals and letting you reuse the same layout code across contexts.
But, to the degree you can live with your set of conditionals and/or repeated code across multiple files, this conditional approach totally works, is native to EE, and is something you can quickly whip out, conceptually and practically.
Also, it's totally fine in EE to have things like News and Events in separate channels, even if they are very similarly structured content. Very different content structures usually warrant different channels, but it's also OK to separate content based on user access requirements, ease of finding things in the control panel, how you want to structure your templates, and other kinds of content "filtering" you might do in your templates.
Alternatively, you could have a News & Events combined channel--e.g., using separate categories or a field to determine which item is news vs events.
Whatever approach you use, there are trade-offs. EE's biggest selling point, imho, is that it provides a bunch of parts you can put together different ways to match your content. But EE's options do mean that you always have multiple choices, that you need to think about this architectural kind of stuff, that there's usually no truly right way to do things, and that you'll probably kick yourself at least once per project for choosing one way over another. But, in the end, you'll make it work :-).