Well, that ended up being much faster than expected. I bought the server on Sunday, received it on Tuesday morning, installed a HTTP server, PHP-fpm, and MySQL, configured the system, moved the site to the new server, and finished everything up by Tuesday night.
So...now that the migration is finished, I need people to run through it and tell me what broke. When we first wrote the site, we made it module so that moving to different servers would be easy. Sadly, when I took over as the sole owner, I have not been as meticulous with my coding...
...so I expect the "newer" parts of the site (things that I wrote after taking over HBrowse) to be potentially broken. That includes things like text search, advance searching, staff pages, aesthetic changes, etc.
Also, some comments on the site's speed and connection would be nice. But I guess, we have to wait for at least the weekend (HBrowse's peak hours) to really stress test the new server.