It seems the site move came off without any major problems. The most problematic thing was broken permalinks for a while. Because this server is running PHP as CGI it won't understand variables in the URL without using mod_rewrite (which I knew but had forgotten). That means I couldn't just keep my old permalink structure which included index.php. As a result, all the backlinks I have to this site became broken and I had to change the permalink structure to exclude index.php.
This was going to be a big issue for me because, needless to say, I wanted to keep my backlinks. After a little searching and learning about mod_rewrite I figured out how to make both permalink structures work. But at some point in the distant future I will disable the old option so I'm changing as many of the backlinks I know about as I can. If your site links to a post on this site, I would appreciate it if you could take "index.php/" out of the link URL.
I still haven't moved the database that houses all the precious content on this site but I plan to this afternoon. Overall, I'm very impressed with Dreamhost (go on, sign up with them and earn me $97). The control panel is much better to work with than any other I've seen. Better than that, I just received my first monthly newsletter from them - the company has a sense of humour, yay!