I'm currently with another hosting company that is also a VDS. When I sign up, is there any way I can migrate my data from the old host to the new host easily and quicker than doing it a file at a time by hand via regular FTP?

rsync

scp -r

rsync over ssh - best of both worlds!

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/ Peter

This will copy the files to the new server, right? It won't delete them off the old server just yet will it?

Correct. None of the proposed methods will remove any files from your source server.

I'd recommend rsync only because it's easier to make sure that the files are copied exactly (file permissions and ownerships included).