There are various combinations of things that can create your desired layout. You could go with Template Overrides for blog category, or Featured Articles layout, but maybe an easier approach is to use Modules for displaying news.
Recommended module: Raxo All-Mode Pro
One of my favorites one in my opinion and of the best Joomla news display modules is the RAXO All-mode Pro. I use it in most similar to yours cases.
Some of its features:
- Easy to use.
- Many options for selecting/filtering what articles to display
- Options to show article metadata like images, author, category, hits, comments
- Good looking premade templates
- Easily create new custom templates / customizations
- Good support.
It's a paid module however, but totally worth it.
Basic steps to Create the desired blog layout using modules
So, either with the module I recommended or an other similar one, you could create 3 module instances, one for each category and you will adjust the each module's instance settings to display articles from those categories you want the way you want.
Now, how to display these modules in your page's main content, to server as blog split in displaying 3 categories?
If your template provides a module position at the location of your main content, you can create a Featured Articles, or category blog Menu item, set the number of leading, intro and links article to 0 and publish your 3 modules in that position for that menu item.
Or Create 1 article, name it something like Blog Home Page (don't put it in any of your blog categories - better set it as uncategorized), in the article's body, load the 3 modules with loadmodule/loadposition (or use the new J3.5. module editor button), create a new single article menu item, and assign this article to it.
You will have your desired layout in minutes and may end up with something like this:
Bonus: Links about Joomla Overrides
Note: I am not affiliated in any way with Raxo module