Okay - general rule with ALL gemstones is to do as I said above, including Tanzanite.
1. Warm tepid water.
2. Soft (preferably baby) toothbrush
3. Tiny amount of washing up liquid
4. Lint free cloth.
Dunk toothbrush in water and put a tiny weenie drop of washing up liquid on the wet brush. Put under water and brush on your hand until you get a gentle foam. Then gently brush over the gem, repeating until the gem is clean. Then clean your brush under water so there is no soap left, shake off excess water and then keep going over the gem until it's free of any water and soapy liquid.
DO NOT EVER WATER-LOG YOUR GEMS - you must not put anything under running water.
I'm not really great with silver as for some reason, my pieces don't seem to tarnish much so I'm hoping that others can help you here. What I do is use a silver cleaning cloth but be careful not to get it on your gemstones. If there are lots of nooks and crannies in the silver shank then you may need a liquid cleaner but beware of getting this anywhere near your stones - make sure you use only a tiny bit and wipe off then repeat.
Hope that helps.