In short

Raman signal rises steeply towards shorter wavelengths. Fluorescence usually does too. The right excitation sits where that trade works out for your sample.

Common starting points: 532 nm for many inorganics, 785 nm for many organics, 1064 nm when fluorescence dominates. Verify on your own material.

For ordinary dispersive Raman, spectrometer resolution, not laser linewidth, sets what you can resolve.