Do not use the free Express Edition (XE) database since it can not be patched. Follow the principle of least privilege, so a user only has access to the resources required. Lock or remove unused users. Use sensible passwords, and do not use the same password for SYS and SYSTEM. There are lots of more advanced things you can do so check these:
Oracle has created Project Lockdown which is a project that via 4 phases helps you to secure your database infrastructure
Hacking and Securing Oracle - A Guide To Oracle Security (24-04-2007) by Pete Finnigan. Scary reading. Contains a very informative list of steps to secure your database.