Generally, switching should be fairly straightforward. Instead of mainPart:SubtractAsync(otherParts) you have to call
local GeometryService = game:GetService("GeometryService")
GeometryService:SubtractAsync(mainPart, otherParts)
The new API has a bunch of cool features that the old API does not have (like the ability to be called in local scripts). Please take a look at the documentation here for more information.
Also, back when we introduced the new API, we provided some example place files. These files are also a great starting point for exploring some of the more advanced features the new API has.