I think the reply doesn't depend on the software you choose but the needs you have.

for simple persistence DBM will often be enought (or DBM::Deep which is very perlish ) , if you plan complicate data structure you may need a SQL interface involving DBI. which you can plug on different DataBase.

as Database itself DBI interface plenty of it. Postgres has a full featured SQL implementation, while MySQL is simple to install and performant. But if you just want to train yourself Postgres is ok as well as SQLite or any DBD you can install on your machine and code with.