The Bitnami Redmine Stack is a very clean and simple way of installing Redmine. It also has the advantage of being well supported and documented. It avoids the hassle of having to install mysql, ruby, rails, subversion, etc individually. However, there are some important things to keep in mind:
Generate a dump file of the old database (if there are multiple databases
(_production, _development, _test), only the production database needs to be migrated):
mysqldump redmine_production > dump_file.sql
Migrate the database:
mysql -u username -p new_database_name < path/to/old_database_dump.sql