In MySQL, how can I delete/flush/clear all the logs that are not necessary?

FLUSH LOGS just closes and reopens log files. If the log files are large, it won't reduce them. If you're on Linux, you can use mv to rename log files while they're in use, and then after FLUSH LOGS, you know that MySQL is writing to a new, small file, and you can remove the old big files.

Binary logs are different. To eliminate old binlogs, use PURGE BINARY LOGS. Make sure your slaves (if any) aren't still using the binary logs. That is, run SHOW SLAVE STATUS to see what binlog file they're working on, and don't purge that file or later files.

Also keep in mind that binlogs are useful for point-in-time recovery in case you need to restore from backups and then reapply binlogs to bring the database up to date. If you need to use binlogs in this manner, don't purge the binlogs that have been written since your last backup.

If you are on amazon RDS, executing this twice will do the trick:

PROMPT> CALL mysql.rds_rotate_slow_log;
PROMPT> CALL mysql.rds_rotate_general_log;

Source: http://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_LogAccess.Concepts.MySQL.html