The existing law provides a number of requirements that non-EU citizens are required to respect even before their arrival in Italy, among which there is also that of an "adequate insurance coverage for medical care and hospitalization" (Art. 39 c.3 Act no. 286/1998 and Directive 01.03.2000 of the Ministry of the Internal Affairs). Non-EU citizens can get this visa entry for study reasons - needed to get permit of stay once you arrive in Italy - only in one of these three ways:
1) release from the local Italian Diplomatic Representation of a consular declaration stating the right to health care under agreement between Italy and the country of origin;
2) drawing up a foreign insurance policy, accompanied by a consular declaration of its validity in Italy, its duration and the forms of assistance provided, that should not lead to limitations or exceptions to the tariffs set for urgent hospitalization for its entire duration;
3) drawing up an insurance policy with agencies or national companies, such as the National Insurance Institute (INA) which provides in agreement with the Ministry of Health a special insurance to cover specific risks (€ 49 for 6 months, € 98 for 12 months); in case of other agencies or other companies different from INA, the policy must be accompanied by a statement of the insurance istitute that specifies the absence of limitations or exceptions to the tariffs for the urgent hospitalization for its entire duration.

Owning the health insurance, whether Italian or stipulated in the country of origin and stamped by the Italian Embassy, you can go to any doctor among those available from the list published on the website of the Lazio Region. If you ever need to see a specialist, you can directly choose which doctor to consult, possibly taking advice from the doctor. You will have to pay the specialist visit  and later seek reimbursement to your insurance company.