How to know and which to use?
If your goal is to convert your 3 gears in the front and 7 gears in the back to a 3 in the front and 9 in the rear this will be quite expensive. A new rear wheel will be required Your current rear wheel uses a screw on freewheel to mount the rear gearset. Nine speed rear wheels use a freehub mounting system with a cassette. You will need a 9 speed rear shifter and derailleur of the same brand, Shimano shifter with a Shimano derailleur or Sram derailleur with Sram shifter. You may need a new front derailleur, some 7 speed front derailleurs can be adjusted to work with narrower 9speed chains but it is trial and error and shifting may not be optimal. If you front shifter is a non indexed type this may actually improve shifting in this mismatch scenario. You will need a 9 speed chain. If you are doing these improvements yourself you will need to add the cost of a chain-tool, freewheel removal tool, cassette install/remove tool and a chaiin whip.