You might have to replace the shifter and the chain, too, besides one or both derailleurs as Blueberry pointed out.

+1 on getting a mechanic to help you with the parts selection and do the installation. Drivetrain parts aren't necessarily interchangeable. Also +1 on at least considering a wider range cassette (which might still necessitate a new rear derailleur and definitely a new chain, but not all the other stuff).