Sweden has a national healthcare system. Policy is dictated by the government, and the healthcare services are run by the regional authorities, local authorities or the municipality.
To access the national healthcare system you should be insured through the National Social Security Department (Försäkringskassan). On their website, you can find an introduction to the Swedish social security system, here.
In the national healthcare system, patients sometimes pay a small out-of-pocket sum for some consultations or treatments or prescriptions. There is some variation in these costs between regions, but within the region a provider cannot charge more than the set rates for the region: you do not need to worry what hospital or doctor you go to. The government caps the amount of co-pays for each patient you can find the current cap, in Swedish, here. If you have paid this amount in a 12 month period, you will get what is called a free card (frikort), which means you will not pay any more co-pays until the period resets. This applies to visits to the doctor or hospital, as well as to prescription pharmaceuticals.
Dental care is not generally free in Sweden, unless you are younger than 23 and registered with Swedish Social Security. The year you turn 24, you start paying for your dental care in almost all regions, but even then you do get a state dental care subsidy. Here you can find information in Swedish about dental care.
A personal number entitles you to social insurance, including health insurance, through the National Social Security Department (Försäkringskassan). Here you can find information about what applies to individuals who are moving to Sweden.
You can ask at the Tax Agency, as soon as you apply for your personal number, about registering with the National Social Security Department. While this is often handled automatically when receiving a personal number, it may happen that the information is not submitted directly from the Tax Office to the National Social Security Department.. You should check that you are insured properly, by contacting the National Social Security Department.
For the period between moving to Sweden and getting your personal number, you should get health insurance for yourself. If you are residing in a EU/EEA country, you should apply for a European Health Insurance Card (sometimes called the blue card) in your country of residence; this card will give you access to healthcare at the same rates as Swedish residents. If you are from outside the EU/EEA, you should look into purchasing travel insurance for the period until you have received your personal number and registered with Försäkringskassan. You should ask your employer about the possibility of covering the cost.
Being registered with the National Social Security Department is also necessary for claiming sick leave benefits and parental leave benefits, as well as for applying for prolongation of your doctoral studies due to leave. The administration staff or HR office at your department can most likely help you with instructions on how to register leave with Försäkringskassa.