There are two natural reserves encompassing the Tatra Mountains, one in Slovakia and the other in Poland. The landscapes and scenery they have to offer are simply wonderful, and at the numerous lakes within their confines there are many beautiful spots in which to relax and/or contemplate the surroundings in peace and tranquility.
There are also various waymarked hiking trails in the reserves, offering something to suit everyone from the vigorous outdoor type to the more casual walker. The easiest trails are located in the lower parts of the mountains; the hikes you can do here are more akin to strolls in the open air. The rest, by contrast, skirt around the mountain summits. These are reserved for the keen hiker however.
I am only familiar with the the lower Slovakian region of the Tatras. I stayed at a hostel in Ždiar and while there took advantage of the local restaurants serving delicious traditional dishes, halušky and pierogi in particular! It was possible to venture off into Poland every day by following the waymarked trails. The scenery was wonderful, and it wasn't unusual to encounter local wildlife along the route.