by Ken Task.
precautions - but I see you've already started ...
Check old sites additional plugins. If you have any, on the new server install compatible versions of same plugins.
Assuming old shared site => new shared site in space/memory and other specs of your slice of the shared server.
Might increase default PHP values for time for a script to run, memory a script can consume ... seen in Site admin menu -> Server -> PHP Info.
Check backup defaults on old server. Make sure users are included.
'spirit of sharing', Ken