The guide here assumes that you are a non-Italian citizen and you are hired on a contract in which you don't pay normal taxes. This is true for most postdocs and PhD students.
Note that registration to Italian public healthcare is always for a full calendar year, from January to December. You would pay for the whole year even if start in December. Therefore, it is only feasible if you sign up in the beginning of the year. Also, know that you cannot register for healthcare before the year has actually started. Be careful if this creates a gap in your coverage.
Expect all officials to only speak Italian. Bring a friend if necessary.
WARNING: It appears that the process has changed since the writing of this guide. If your fee is paid for by the department, they may now pay it directly instead of sending you to the postoffice. Please update the steps accordingly so we avoid difficult double payments.
The process of signing up for the city (i.e., the residence) is linked to the process of signing up for health care so these two processes must be done in parallel. This is because payment of the health care fees is required in order to complete the registration for the city, which is in turn is required in order to sign up for the health care. Here you can encounter a circular dependency. Follow these steps carefully to break the loop:
For extra credit, about 20 days after completing your residence registration you can apply to receive an Italian ID card.
Yes, it's not a one-time-only problem. You will have to renew your healthcare registration each year. Fortunately, repeat registrations are simplified by your residency already being sorted out.
Here I can only speak for people with a contract with the University.
You have to go to the INPS website (<https://serviziweb2.inps.it>) and register for an account with a password. There are several ways to register, I was told to use the "PIN" authentification. The login is your Italian Tax Code, and the password is what is called the PIN.
In order to get the PIN, you have to call +396164164 to fix an appointment at INPS Padova (via Delù, 3). At the appointment they will ask you for your Italian Tax Code, and give you the first part of the PIN by email and the second part of the PIN by hand.
Then, go to the INPS website (<https://serviziweb2.inps.it>) thanks to your PIN and put your information to activate your account.
Once activated, apply for a "gestione separata" (Iscrizione dei lavoratori parasubordinati alla Gestione Separata) on their website by filling out a very short form. Note that the date to put in "modulo parasubordinato" is the starting date of your contract. Once the application is successful, download the ticket and send it to the department's Secretary.
Italians generally have an official, digital ID known as an SPID. This can be used to complete official procedures entirely online. However, if all you want to do is a postdoc and are already registered in the city, then you might not need this. The same is true for the official, Italian mailbox used for official communications, the PEC.