You should definitely use Maintenance Plan functionality to do your backups. You can easily configure it to create new backup file for each run (it will append full date & time to the end of file name, e.g.
One of the possible tasks of Maintenance plan is Maintenance Cleanup Task that is used to delete unwanted *.bak & *.trn (or any other) files. The accuracy (time interval that defines if file can be deleted) is hour/day/week/month/year.
You can insert
Execute T-SQL Statement Task to be run before such Maintenance Cleanup Task. There (in Execute T-SQL Statement Task) you execute external command to copy/compress your backups (e.g.
EXEC master.dbo.xp_cmdshell 'C:\path\to\compress.bat).
I'm doing this myself on our servers where SQL Server 2005 is still used (using WinRAR to compress). SQL Server 2008 has support for compressing backup files build-in (WinRAR does better compression but takes considerably longer time).