People ask this as one question and it's really five. Taken separately, the answers are more useful.
Crime
Tanzania is a stable country and violent crime against visitors is uncommon. Opportunistic theft in towns is the realistic risk: phone snatching, bag grabs and pickpocketing in Arusha, Moshi, Dar es Salaam and Stone Town, like any city anywhere. Don't walk around at night with a phone out, use taxis your hotel calls rather than ones that call to you, leave the good watch at home, and keep your passport in the safe with a photograph of it on your phone. In the parks and the camps, this is essentially a non-issue.
The animals
Safer than the photographs make it look, as long as you behave. In the vehicle you're a shape rather than a person, which is why you stay seated and inside it. In unfenced camps you'll be walked to your tent after dark by a member of staff, and that isn't theatre: it's because buffalo and hippo wander through, and both are more dangerous than the cats. Follow the camp's rules and the risk is genuinely small.
The roads
Statistically, the real one. East African road traffic is the most likely thing to hurt a visitor, which is why the quality of your vehicle and the hours your driver is asked to do actually matter. Long transfers at the end of a day, or a schedule that needs the driver to hurry, are the bits to push back on.
Health
Malaria is present in most of the country and is the thing to take seriously. Stomach upsets are common and mild. Good hospitals exist in Arusha, Moshi and Dar, and nowhere near a park, which is why the evacuation clause in your insurance matters more here than in most places. The detail is in health and vaccinations.
Politics and the news
Tanzania has been politically stable for a long time and tourism continues through election periods. Your own government publishes travel advice for Tanzania and that's the source to check before you fly, rather than a tour operator's reassurance, including ours.
Travelling as a woman, and other specifics
Solo women travel here routinely and generally report it as straightforward. Dress is more conservative than Europe, particularly in Zanzibar and in towns, and covering shoulders and knees attracts less attention. Same-sex relationships are criminalised under Tanzanian law, which is a real consideration for LGBT travellers; discretion in public is the practical advice most travellers are given, and your own government's advice page covers it properly.
What we'd actually tell a friend
Watch your bag in town, wear a seatbelt, take the malaria tablets, listen to the camp staff after dark, and then stop worrying and enjoy it.