In the village you can park your car in one of the paid parking lots, for example near the Grand Hotel (P; GPS: 49.141327, 20.222184), but you have to reckon with the fact that the cost would be around 5-6€ per day. Therefore, it is acceptable when you plan to go the hiking trails in the mountains just for the whole day. In other case, when you want to visit just the village and you need only 1 or 2 hours, you should look for another place on one of the side streets to avoid such charges.
You may walk down the streets, where you can find buildings with distinctive facades. Near Grand Hotel you will see interesting church of the Immaculate Conception (A; GPS: 49.141213, 20.220967) - unfortunately it's closed quite often. In addition, you can visit the station of the narrow gauge railway (B; GPS: 49.139288, 20.222602) connecting Poprad, Tatranská Lomnica and Štrbské Pleso.
More experienced tourists can walk the hiking trails leading from the Starý Smokovec to High Tatras. For example, a good map is available on the website http://mapy.cz.