Healthcare of Lithuania is of a formidable standard with numbers of doctors per 1000 people larger than in most Western societies. Hospitals are well-equipped to perform even the most difficult surgeries. The doctors are well-trained and sought-after by Western hospitals.
For Lithuanians and people of the European Union, most medical services are free of charge. However, corruption is rampant, meaning that a person with relationships among doctors (or a bribe) may get a preferential treatment bypassing the queues (which may get long, depending on location and procedure). This used to be the norm in the Soviet Union when all goods and services were in theory equal-to-all but in reality depended on bribes and relationships; today it declines.
This, as well as Soviet "patient-is-always-wrong" attitudes in some public hospitals (also declining), makes a part of the population to pay full price at the private clinics (even though they are still subjected to massive compulsory public healthcare taxes). This heavily depends on a specialist: nearly everybody visits a private dentist or gynecologist just as nearly everybody uses the public hospitals for major surgeries. For a foreigner private hospitals may be less of a hassle in all cases especially if one has an insurance coverage for them. Even without it, many procedures may be cheaper in Lithuania than the West (dentistry, plastic surgery). Lithuanian emigrants come back home to perform them.
If you choose public hospitals the best ones (and the largest) are in Vilnius and Kaunas.
Lithuania has a wide range of health resorts and spas, especially in Druskininkai resort.
There is generally no need to get any vaccination before going to Lithuania. Major infectious diseases such as Hepatitis B, Hepatitis C or HIV are extremely rare.