You can install the new theme and preview how it would look before changing it.
See https://en.support.wordpress.com/themes/#preview-themes for more info.
Once you have tested that it works, you can then activate the new theme. I would keep the old theme for a few weeks until you’re confident that there is nothing else needed, then take a backup of old theme before deleting it.
I’d also review the current theme’s functions.php file for any customised code that you may want to copy in to the new theme’s functions.php file or write a plugin to make it compatible across all themes.
As a final note though in case not obvious – install the new theme first before deleting the current/active theme.