Szentendre on the Danube north of Budapest is a baroque arts town with the Szamos marzipan museum, Hungarian bistros on the riverfront and a Saturday craft market, half-hour HEV away.
Tip: HEV from Batthyany ter on the Buda side; town is walkable, no car needed.
Visegrad on the Danube Bend is the medieval royal palace town where Hungary, Bohemia and Poland summit-summoned in 1335, with riverside Hungarian inns and a citadel above.
Tip: Train from Nyugati to Nagymaros, ferry crosses to Visegrad in 5 min; come back by train via Szob.
Lake Balaton is the Hungarian Riviera, with Olaszrizling whites from the Badacsony vineyards and grilled fogas at lakeside restaurants in Balatonfured and Tihany.
Tip: Balatonfured for the lake's spa town; Tihany ferry from Balatonfured port runs Apr-Oct.
Pannonhalma is the 1000-year-old Benedictine archabbey north-west of Budapest where the monks pour their own wines and run a UNESCO-protected cellar, a quiet day trip.
Tip: Train to Györ then bus 23 min to Pannonhalma village; wine cellar tours run Tue to Sun.
Eger is Hungary's most famous wine town, the home of Bull's Blood (Egri Bikavér) and the Valley of the Beautiful Women, with 200 volcanic-rock cellars open to the public.
Tip: From Eger station, walk 10-15 min into town or head straight to Szepasszony-volgy for cellar tastings.
Tokaj is the world's first delimited wine region, a UNESCO heritage zone north-east of Budapest known for Tokaji Aszú sweet wines and serious dry Furmint from villages like Mád.
Tip: Tokaj town is 3 hours by direct train; Mád and Tarcal villages need a taxi from Tokaj station.