Indeed you WILL be able to go directly from 3.0.14 to 3.2.3.
But just speaking from my own recent 3.0.14 to 3.2.2 project experience, do NOT expect success on the first try, unless you just happen to have a particularly easy board or with zero mods installed and only a few thousand posts. Particularly if you had mods installed, and intend to continue using the data from those mods with later versions of the extensions made for 3.2.x, you just may end up with a few questions or issues to work through that will be less stressful if your production board isn't down or malfunctioning at the time.
I would highly recommend leaving the production board alone, and using a backup of the database and of the file system to setup your current board in an alternate testing environment. Where you can attempt upgrades and work though any issues, and reset back to the starting state multiple times as needed, until you determine all the issues, and the steps needed to address or avoid them. And only once determined and tested, will you then come back and attempt those same steps against the actual production board.
As to what that "alternate environment" is, that's up to you. If your current hosting service already easily allows you to have multiple pbpBB instances, you might set it up on a subdomain of your existing hosting account. Or if you want to keep it completely separate, and/or want to run with a different PHP version than your current hosting account is set for, you might purchase a temporary alternate hosting account just for this purpose. Or you might setup a local Windows environment for testing, as described in Installing and Setting Up Your Own Web Server