The Great Market Hall (Vámház körút 1-3; 00 36 1 366 3300) is a bustling place to shake the sleep from your eyes. Built in 1897 and topped with multi-coloured roof tiles, its design is a wonder in itself. Beneath its girders, stall-holders hawk fresh produce to locals and bags of powdered paprika and lace tablecloths to tourists; if you missed breakfast, there are open kitchens on the first floor selling buffet-style food and snacks.
Some of the local dishes you should look out for are hortobágyi palacsinta (pancakes stuffed with minced meat and baked in a paprika sauce), gulyásleves (the classic goulash soup) or lángos (flattened, fried dough ladled with garlic sauce, sour cream and grated cheese).
Cross the road to join Váci utca, the pedestrianized artery that runs through the heart of the Belváros (Downtown), lined with boutiques – but beware the over-priced gift shops. It's a vibrant street of buskers, history (look out for no 9, where an 11-year-old Franz Liszt gave a performance in 1823) and the occasional intriguing statue (a favourite is the 'Fisher Girl' in Kristóf tér). After a kilometer, you'll emerge into the grand Vörösmarty tér, where it would be a crime not to stop for a cake at Gerbeaud (Vörösmarty tér 7-8; 00 36 1 429 9000), the most famous café in Hungary.
Head out of the square's top right corner towards Erzsébet tér, where you could take a spin on the Budapest Eye (00 36 70 636 0629) with its views across the rooftops. It's a short hop from here to the domed St Stephen's Basilica (Szent István tér 1; 00 36 1 311 0839), which also offers splendid views from its gallery (if you can brave the 300 steps) – and a casket containing the mummified right hand of St Stephen, the country's 11th-century founder.
The afternoon is all about Buda. After a spot of lunch at Bistro Fine (Andrássy út 8; 00 36 1 611 7090), a relaxed restaurant a short walk away on Andrássy út, settle your stomach with a stroll to the Danube and across the iconic Chain Bridge. Join the funicular railway (the lower carriage has the clearest views) for a trundle up to the medieval quarter. With its cobbled alleys and pastel-coloured burghers' houses, the Castle District oozes romance (although it can also get crowded).
Turn left into the palace complex, home to the absorbing, vast National Gallery (Szent György tér 2; 00 36 06 20 439 7325), which holds over 100,000 works of Hungarian art ranging from medieval stone carvings to dramatic canvases by 19th-century Romanticists like Károly Lotz.
When you emerge, don't miss the Mátyás Well in a courtyard behind the palace – a bronze fountain showing Hungary's great Renaissance king hunting in a forest. The same king was twice married in Mátyás Church (Szentháromság tér 2; 00 36 1 488 7716), which stands a few hundred meters away, its interior is an eye-smacking riot of color.
Start the day with a walk along Andrássy út and you'll understand why Budapest has been called the 'Paris of the East'. This elegant boulevard was the brainchild of 19th-century nobleman Count Gyula Andrássy, who wanted the city to have its very own Champs-Elysées. As you walk away from Bajcsy-Zsilinszky út – neo-classical mansions forming a guard of honour either side – you'll pass the beautiful State Opera House (Andrássy út 22; 00 36 1 81 47 100) and the many café-bars around Liszt Ferenc tér.
Standing just beyond Oktogon, the Terror Háza (Andrássy út 60; 00 36 1 374 2600) is well-named. This was the headquarters for first the Nazis and then the Communist secret police – a place of brutal interrogation, torture and execution. Today the building contains a stylized museum focused upon Hungary's terror regimes that's as fascinating as it is chilling.
Andrássy út reaches a dramatic climax in Heroes' Square, laid out in 1896 to mark the 1,000th anniversary of the arrival of the ancestral Magyar tribes. At its center is a 36m-tall column topped with the Archangel Gabriel, while behind are colonnades bearing statues of some of the country's greatest leaders.
Gundel (Gundel Károly út 4; 00 36 1 889 8111) is the place to eat this afternoon. It has an illustrious history and a sumptuous dining room, but for day-time visits there is also a less formal terrace and a good-value set lunch.
After that, City Park has more than enough to fill a few hours. You can go boating on its lake (Olof Palme sétány 5; 00 36 20 261 5209), which lies in the shadow of the quirky Vajdahunyad Castle (00 36 1 422 0765), a fairytale hotchpotch of a building that incorporates over 20 different Hungarian architectural styles. Nearby, the Museum of Fine Arts (Dózsa György út 41; 00 36 1 302 1805) in Heroes' Square has some superb exhibitions.