Tanzanite is prone to fracture, so definitely avoid mechanical cleaning like sonic baths. It's best to use a soft brush, mild detergent, and warm water as recommended in our cleaning guide.
As for care, store your tanzanite separately from other gems to avoid scratching it, and try to avoid bumping it against anything, since it can fracture easily. Having the stone in a protective setting, so that the edges can't catch or snag, will help avoid fracture.
If you want to know more about your gem, there's tons of information on tanzanite in the tanzanite gem listing and buying guide.
Geologist and IGS Team Member