PostgreSQL and mySQL are both free, but keep in mind that you do indeed get what you pay for.
If all you want to do is build tables, fill them with rows and fetch them back, either of these two is your best bet.
If youi plan to write stored procedures, functions, and triggers, both PostgreSQL and mySQL are POOR choices beccause their programming functionality is either brand new (bugs) or non-existent.
If your database server is a Windows box, I would STRONGLY suggest SQL Server
Of all of the DB packages, I prefer Oracle as do many people. It is moderately difficult to install on a Linux/Unix machine, but it has the BEST programming functionality with PL/SQL. It's performance is top notch.
MS Access is not meant for large, complex applications and should not even be considered