Last week I was asked about how to handle current users who may be confused or frustrated by a redesigned site that has a new layout, navigation, etc. Ideally the new site will be easier to use for all concerned but for people who learned the old site, there will still be a learning curve.
From my perspective, the best path to resolve this issue really depends on what your current audiences come to the site for and if you will continue to support those same outcomes with the new one.
If you have people who regularly come to your site to complete a specific task or get a piece of content, and you will continue to support those outcomes, you can make sure to still facilitate those actions even in a redesigned site. Ideally the new design will facilitate those well but you can also make a set of custom pages for specific audience segments that guide them to the new location of these items. These pages can be promoted directly or provide as a highlighted help tool for a period of time after the launch of the new site.
On the other hand, if the site will serve entirely new outcomes, then your audiences will have no choice but to learn the site anew and there is not much you can do to avoid that.