It depends on the difficulty of the trails you want to visit.
If you like hiking up the high mountains the national symbol Krivan is something for you... the hill itself is very photogenic.
The traditional, easy but very nice hike leads from Strbske pleso to Popradske pleso and symbolic cemetery around several huts ups to the the Rysy mountain (2499 m on the border with Poland) or to the Velke Hincovo pleso and Kopovsky stit.
Great views of the High Tatras are also from small Tatras- Kralova hola (1946 m) just 30 km from Poprad. about 9 years ago