If you are moving your library to a new machine or even a new drive, these instructions may be helpful.
You can use Windows to move the files to a new drive or across the network. This will require also changing the drive letter manually (see Library Files below).
If you're just moving to a new drive on the same machine, you can save some trouble (and eliminate the next step) by having MC move the files. This is done by selecting the files in MC, then right clicking on them and choosing Library Tools/Rename, Move, and copy files. This will also change the drive letter in the library so you can skip the next step. As always, try a few files before you do very many, and verify your results.
MC makes backups of the database itself automatically, but you can also make one manually from File/Library. On the new machine, you can restore.
Changing the drive letter
The same Library Tool from above allows you to replace any string. In this case, replace C: with F:, for example.