Balatonfüred is the main town just northeast of Tihany and the most useful transport gateway for the peninsula. It has a more formal, old-spa atmosphere than the smaller villages around it, with promenades, villas, and cafés facing the lake.
It makes sense to combine the two places: Balatonfüred for arrival, dinner, or a longer lakeside stroll; Tihany for the abbey and peninsula landscape. Buses and taxis connect them quickly in season.