Check App Reviews, Compatibility and Install Updates
Many Mac apps will work with macOS Mojave as soon as you install the update. That's one of the benefits of a lengthy beta process, which allows developers to test and prepare an upgrade with support for Mojave and that can take advantage of new features.
That said, not all apps will be ready to go on day one. We typically see this being an issue on specialized apps, niche apps and apps from smaller teams -- but we've also seen major compatibility issues from Microsoft and other big names.
Take time to look on the support pages for your essential software and to look into the reviews on the Mac App store to see what other users are saying. If you have a space machine, use it to test compatibility.
Here's an unofficial list of apps that work and don't work on macOS Mojave. This is less of an issue with macOS 10.14.3, but if you are coming to macOS Mojave for the first time with this update, you should still research.