Moving the entire site to a new server went well. The only issue that I've seen is that any attachments made within 24 hours of the move were lost. They will just have to be re-attached by the person who made the post/thread.
There was also a few hours while the DNS propogation took place where some people were on the old site and some were on the new site. If you made a post, or were reading a post, and it's now gone, it was because the site was copied then installed elswhere, and there was that small amount of time where the was an overlap.
Everything is good to go. Thanks for your patience!