Probably the best way of getting around the southern side of the High Tatras is using the Tatra Electric Railway services. This company provides the main public transportation connecting major Tatra resorts with Poprad-Tatry. All trains get to Smokovce, the center of the Tatras, but if you want to follow to Tatranska Lomnica, you must change here.
Cog-railway connects Strbske Pleso with Tatranska Strba, where you can interchange for regular trains connecting the Tatras with all major Slovak cities. Please note that IC trains do not stop in Tatranska Strba.
There are several funicular lines that you could consider using, while staying in the Tatras. Probably the most popular Tatra line is the funicular from Stary Smokovec (valley station is just behind the Grand Hotel) to Hrebienok. From here you may descend to the Cold Creek Waterfalls (Studenovodske vodopady), which is one of the most pleasurable walk in the Tatras.
Taking a gondola to the Lomnicky Peak, the second highest in the Tatras, is also an option. Leave from Tatranska Lomnica to Skalnate pleso, a typical Tatra glacial lake, and from there continue to the Peak, offering excellent views of the High Tatras. From here you may also see the panorama of the Belianske Tatras and the Low Tatras. Finally, you should not miss a popular chair lift from Strbske Pleaso the Predne Solisko and descend down walking, enjoying a nice Tatra hike.
Buses commute between major Tatra towns, some also cross to Poland to Nowy Targ and Zakopane. Poprad and Stary Smokovec and two major hubs of the bus transportation in the Tatra region. There are many buses from Zakopane to Cracow.
Using your own car in the Tatras is easy, though during the winter time, you may require chains and driving might be more difficult due to snow. Another problem that appears all year round are the high parking fees and sometimes while visiting a tourist attraction for an hour, a local village or "entrepreneur" leasing the parking lot will charge you for a full day.
There is a comfortable D1 motorway on the Slovak side of the Tatras. In the High Tatras the main road is called Cesta slobody (the Freedom Road).