I have to say that it gave me a lasting impression of what a company can lose when handover fails.
Luckily, I've also had some good handovers and the best way I've found to do it would be to book the whole week as a workshop. Nothing the outgoing staff member can do is more important than the handover and it can often create a level of goodwill in that you are asking for their assistance and making them realise how important they have been.
However, there are also some rules to the week. Some apply to you - you need to check what you are being told. Anything that starts to look hinky or just plain wrong needs to be constructively challenged. If it still doesn't add up after a challenge and you know they are lying then you need to get them on garden leave as soon as you have the keys and passwords.
My approach for general sysadmin has been to try to understand the systems from the ground up very quickly and I've found it useful to have the following as general headings:
1) Passwords - where they are, how they are kept, what policies are in place? Generally find out how it has been managed in the past. Most important - verify them.
2) Network diagrams - use network scanning and mapping tools to verify what you are finding
3) Infrastructure services - understand the setup for anything important to the infrastructure of the systems. Things like DNS/DHCP/NIS/Kerberos/Pam/LDAP/AD/Certificate Authorities/Identity Management/etc/etc.
4) Storage services - SANs/Makes/models/Where to find support contracts/BACKUPs/Data replications/File stores/etc.
5) Core end user services - File/Print/Core Databases/Core Apps.
6) Cloud services/domain registration accounts/3rd party supplier access
There will always be more to find out but hopefully having a list of what you need can stop your company wasting a lot of time and money in having to rebuild what it can't support.