Out of the box, when choosing options, the images and the price will change. **But not SKU.**More information about which elements on the page change when options are selected can be found here: Magento\ConfigurableProduct\Block\Product\View\Type\Configurable::getJsonConfig().
In order to get the SKU of the selected product on frontend, you can create a Plugin (Interceptor) for Magento\ConfigurableProduct\Block\Product\View\Type\Configurable::getJsonConfig() where you need to add missing SKU for each simple product.
Switching SKU when selecting an option can be implemented in vendor/magento/module-swatches/view/base/web/js/swatch-renderer.js and vendor/magento/module-configurable-product/view/frontend/web/js/configurable.js depending on the type of attribute (Visual Swatch, Text Swatch, Dropdown).
I made free VCT Change SKU Dynamically module that solve this problem.
