Updating to latest version

Hello all! I’ve been put in charge of a very old WordPress install running 3.0.1. I’m looking to update it to the latest version because of security issues and of course the ability to modernize the site with new plugins.

Are there any issues I should be aware of in updating from such an old version to the new? Specifically, issues relating to the database and themes, as access to the database is a bit restricted (school server) and we are running a custom theme.

Thanks!