With macOS, you can move all the information from your old PC to your new Mac. I haven't tried using Access on OS X directly, but from looking at Microsoft Access in AppDB (a compatibility list for Wine), it could work on Wine, which runs some Windows software directly on the. Swift Publisher does the trick for nearly every use case, and MarsEdit effortlessly takes care of your web presence. Lots of apps could be called a Microsoft Publisher equivalent for Mac. Moving from your PC to your new Mac? Consider it done.
