I figured it wouldn't be too hard to 'install' Apache and MySQL on a server, but I assumed that it would be pretty hard to then actually get it working as described above for someone like me who doesn't know anything about Apache and MySQL. Maybe I'm wrong and there are some good tutorials out there.
Sounds like a good opportunity to learn. You can even test everything locally, without the Arduino knowing what you are doing. When the server works from a browser, then the Arduino can connect to it, and make GET requests, too. The GET request that the Arduino and the browser make are, under the covers, identical. The PHP script parses the data and INSERTs that data in the MySQL database.
MySQL can time stamp the entries, so that the Arduino/browser/user doesn't need to.