Hiking is one of Tihany's main attractions; there's a good map outlining the trails in the center of the village. Following the Green Trail northeast of the village centre for an hour will bring you to the Russian Well (Oroszkút) and the ruins of the Old Castle (Óvár) at 219m, where Russian Orthodox monks, brought to Tihany by Andrew I, hollowed out cells in the soft basalt walls.
The 232m-high Csúcs-hegy (Csúcs Hill), with panoramic views of Lake Balaton, is about two hours west of the church via the Red Trail. From here you can join up with the Yellow Trail originating in Tihanyi-rév, which will lead you north to the ruins of the 13th-century Apáti Church (Ápáti templom) and to Rte 71. From the church, it's possible to follow the Yellow Trail south till it crosses the Blue Trail near Aranyház, a series of geyser cones formed by warm-water springs and resembling (somewhat) a 'Golden Horse'. From here, you can take the Blue Trail north to the Inner Lake and on to the town centre.