That depends on the tool, but usually, the Project Manager is the person who “gives a tour” of the new tool and coaches the client on using it.
The most effective way is to let the client accomplish a task or a process in a given tool – nothing beats practical experience.
For more sophisticated procedures, we might want to document the process in a written tutorial or a screencast.
If the tool is specific to QA or dev work, then people from these roles would take responsibility for walking the client through the tool.