On these pages you will find information about the rules and precautions you should take into account while living in Tartu, as well as how to prevent accidents. There is also a section on general health information. You can also review the orientation presentations on safety issues and the health care system in Estonia on this page. The Welcome Web’s safety page can be found here.
For larger international safety issues, check the website of your home country's foreign ministry/foreign office to see whether it provides any specific safety recommendations. Also consider registering with your home country as an international traveler if your country has such a program. An example is the Estonia Smart Traveller website (in Estonian but it is possible to translate the page using something like the translate Chrome extension).
Keep in mind that while there are a few different departments responsible for ensuring the safety of people living in Estonia, you should dial 112 if you have an emergency of any sort!