Budapest is the obvious starting point and genuinely deserves its reputation. The city combines grand Habsburg architecture, a superb riverfront, excellent museums, a world-class thermal bath culture, and some of the best nightlife in Central Europe (the ruin bars of the Jewish Quarter are unlike anything elsewhere). Give it at least three days.
Lake Balaton is Central Europe's largest lake and Hungary's answer to a seaside holiday. The south shore is flat, busy, and hedonistic in summer; the north shore, with its volcanic hills, vineyards, and the Tihany Peninsula, is more interesting. Come outside July and August to avoid the crowds.
Eger is one of Hungary's most appealing provincial cities — a well-preserved Baroque town with a medieval castle, the country's only surviving Turkish minaret, and a valley of wine cellars known as the Valley of Beautiful Women. The local red, Egri Bikavér (Bull's Blood), is worth seeking out.
Tokaj and the surrounding wine region produce some of the most distinctive wines in the world. Tokaji Aszú, made from botrytised grapes, has been prized by European royalty for centuries. Visit in October during the harvest and you can taste directly from the cellars.
The Danube Bend (Dunakanyar) — the stretch of river north of Budapest where the Danube loops sharply through forested hills — makes an easy day trip or weekend escape. Szentendre, Visegrád, and Esztergom are the main stops: a Serbian artists' town, a ruined medieval royal palace, and a basilica that dominates the horizon for miles.
Pécs in southern Hungary has an unusually layered history — Roman ruins, Ottoman mosques repurposed as churches, and a thriving arts scene. It was a European Capital of Culture in 2010 and has a relaxed, Mediterranean-feeling atmosphere rare in Hungary.
Hortobágy National Park is a UNESCO-listed expanse of puszta (great plain) — flat, windswept grasslands with grey cattle, racka sheep, and the traditional horsemen (csikós) who have worked this landscape for centuries. It's a genuinely different Hungary from the cities and wine regions.
Hollókő is a UNESCO-protected village in the Northern Uplands, preserved as an example of traditional Palóc architecture. Small and sometimes over-visited on holidays, but worth it out of season.