Compress backup file using T-SQL
Backup compression was introduced in SQL 2008 Enterprise, and in SQL2008R2 and later, added to Standard Edition.  

When creating a backup, you can specify the WITH COMPRESSION keyword, which will ensure that the database backup size is compressed to approximately a similar size as a zipped 'normal' backup file.  

For SQL2005 or older, the best way really (other than using a specific tool like RedGate) is to ensure that xp_cmdshell is enabled on the instances, and then use a command line to compact using for e.g. WinRar.  

I use a cmd file that looks something like this: 

@echo off
Set "winrarPath=C:\Program Files\WinRAR"
"%winrarPath%\winrar.exe" a -r "ZippedBackup.zip" "BackupFile.BAK" 


You can then execute this cmd file from your 2000 / 2005 instances.  You can also play around with passing %1-type variables to the cmd file, if your filenames are not generic.