'This place is like heaven for Hungarians," says our guide Gabor Torocsik, as our railway carriage trundles up the narrow-gauge line into the hills above Szilvasvarad. "To us, the hills are something magic."
The forested slopes would not strike the average mountain-dweller as anything extraordinary. But all things are relative, and it seems that Hungarians – whose landscapes tend towards the pancake-flat – go mad for their hills. Certainly the Bukk National Park, near Miskolc, in north-east Hungary, is a lovely day out. We hop off at the end of the line and stroll down the trail through stands of oak and beech, waterfalls and trout ponds glinting through the green.
I detect a certain nostalgia in this yearning for the uplands. On an ancient map at park HQ, Gabor traces for us the great expanse of the Austro-Hungarian Empire, before the 1920 Treaty of Trianon gave away 72 per cent of Hungary to the neighbours.
"You saw those hills as we drove in?" He gestures beyond the forests towards the border with Slovakia. "All that used to be Hungary."
Eger's impressive castle is a monument to past glories: it was behind these walls in 1552 that some 2,000 gallant townspeople held out against an 80,000-strong Ottoman army. Gabor explains how the ramparts are now an important symbol of national pride.
My daughter is not overly concerned about the slings and arrows of Hungary's historical fortunes. To any Harry Potter aficionado a castle is a castle, and this one has hidden cannons, tunnels, and a noisy replica of the defenders' ingenious "firing machine": a wine barrel that they stuffed with gunpowder and bullets and sent down the slopes of the castle to rout the invaders.
Eger and its surroundings offer treats galore for a nine-year-old. First, there are horses: our lunch at Szilvasvarad overlooks a paddock of Lipizzaners, and afterwards my daughter takes the reins of a horse and cart as we trundle through the forest. Then the sweets: Eger's confectionery museum houses everything from ballet shoes to bibles, all created from sugar paste and tempera glaze by confectioner Lajos Kopcsik.
In Eger's Szepasszony Valley, "the valley of the beautiful women", the goodies are more consumable. Here, the soft volcanic tufa that underlies much of the town is honeycombed with 130km of wine cellars. In the chilled vaults of cellar 36, local vintner Tamas Sike treats us to a tasting. "We cellar-keepers know three types of wine," he tells me, as we progress from a 2008 bikaver (bull's blood) to a syrupy 2009 late harvest. By glass number nine, I am laughing at pretty much anything he says, and have begun to acquire the rosé-tinted spectacles that clearly explain the district's name.
The next morning, leaving Eger, I realise we haven't spied another British tourist since the baggage carousel at Budapest airport. Our first stop was in pretty Szentendre, just half-an-hour's drive from the capital. That evening, as we tucked into chilled sour-cherry soup and spicy goulash at the Aranysarkany restaurant, the city-breakers surprised us by their absence. But this was nothing unusual: according to Gabor, 95 per cent of Brits never make it beyond Budapest.
So it is with a smug sensation of having the country to ourselves that we head south-east towards Hortobagy National Park, in the heart of the Puszta. This region is quintessential rural Hungary, land of nodding water pumps and sun-baked steppes. But we arrive to find that days of unseasonal rain have flooded the roads. My ambition to spot a great bustard – the huge bird, native to these grasslands, that is Hungary's best-known feathered attraction – is thwarted.
The rain has not washed away all the birds. Storks and egrets make the sodden landscape appear more Okavango than Europe. I soak up the sense of space as we drive east across the plain to Tuba Tanya, our guesthouse on the park's eastern boundary. Here, we find homespun hospitality and traditional cuisine. Much of the latter comes from the farm, where the livestock includes Hungarian breeds such as mangalitza pigs – prized for their cholesterol-free pork. My daughter's favourites are the puli dogs, so smothered in shaggy dreadlocks that, without the panting pink tongue, you can't tell one end from the other.
Tuba Tanya is close to the spa town of Hajduszoboszlo, home to one of Europe's largest aqua parks. Hungarians love their spas – an Ottoman legacy that nobody minds – and so, as rain continues to lash the Puszta, we brave the steam baths and bubbling hot tubs for an afternoon splash. While my daughter joins the shrieking youngsters on the water slides, I sink into a steamy cavern, where bathers soak like cave salamanders in amphibious torpor.
Clear skies the next day bring a chance to explore Lake Tisza, created in 1973 as a flood defence for the Hortobagy. A 120km raised embankment provides a perfect cycle track. We pedal our rented bikes through meadows thick with wildflowers. After lunch in the lakeside town of Poroszlo, boatman Lajos Szabo ferries us around the lake.
Birds are everywhere: whiskered terns fluttering over the open water; squacco herons stalking the floating water chestnut; pygmy cormorants arranged on a drowned tree. But still no bustards.
Lake Tisza is dwarfed by Lake Balaton, south-west of Budapest and our next stop. The weather is back on track, and it is clear why this lake – the size of the Isle of Man – has flourished as Hungary's summer playground. We ignore the signs to the popular southern shore resorts, however, and head around the northern shore to the Kali Basin, where an undulating landscape of meadows, vineyards and copses seems to strike a perfect balance between Hungary's northern uplands and eastern plains.
Our base is Sarffy House, in the tiny village of Dorgicse. Here, Tamas Giebiser and Kati Sipeki have converted a 19th-century residence into a collection of B&B apartments. The décor and detail are all you might expect from a designer couple from Budapest. The pair have thrown themselves into rural life, baking rough-hewn bread from a huge clay oven in the garden and serving homemade goats' cheeses for breakfast in the converted barn. Add local artists and musicians who drop by, and the result is a place with a distinctly Tuscan ambience.
For three days, we laze around Sarffy House, exploring the farmland, dozing in a hammock beneath the walnut tree and – when we get our act together – following Tamas's directions to local attractions. Top for us are the discreet beaches and towering monastery of the forested Tihany Peninsula – which juts out into Lake Balaton – and the "sea of stones" at Kirandulas, a perfect rock playground for children.
Our fortnight ends in Hungary's north-west corner, where the town of Sopron is closer to Vienna than Budapest. The opulence of nearby Ezterhazy Palace – Haydn's home for 24 years – reflects the former empire's "Austro" side. We lunch at the Raspi restaurant, with a different estate wine for each course. Then we take to a canoe and explore Lake Ferto. Water snakes wriggle across our bows and kingfishers zip along reed corridors, as guide Balazs Molnar explains how this wetland stretches into neighbouring Austria.
It is thanks to a tip-off from Balazs, a fellow birdwatcher, that on our last evening I find myself up a nearby observation tower scanning a sea of grass. Sure enough, two telltale long necks soon pop up, followed by a laborious flapping as two enormous birds take flight. Great bustards. At last.
There is just one thing Balazs didn't mention, though. To reach my bustard site, I have inadvertently driven 5km over the border into Austria. But it still counts, I tell myself.
After all, all this used to be Hungary.
Travel essentials: Hungary
* The writer travelled as a guest of the Hungarian Tourist Board (00800 36 00 00 00; hungary.com).
* Budapest airport is served by BA (0844 493 0787; ba.com) from Heathrow, and by a Malev (0844 482 2360; malev.com) code-share flight from Gatwick and Manchester. Wizzair (0906 959 0002; wizzair.com) and easyJet (0843 104 5000; easyJet.com) fly from Luton, and Jet2 (0871 226 1737; jet2.com) from Edinburgh and Manchester.
* Royal Club Hotel, Visegrad, Danube Bend region (00 36 26 597 100; royalclubhotel.hu). Doubles from 28,210 forint (HUF 28,210/£82), including breakfast.
* Hotel Villa Volgy, Eger (00 36 36 321 664; hotelvillavolgy.hu). Doubles from HUF19,900 (£58), including breakfast.
* Tuba Tanya, Hortobagy region (00 36 30 9 584 550; tubatanya.hu). Doubles from HUF15,000 (£44), including breakfast.
* Kormoran Apartments, Lake Tisza (00 36 59 350 350; kormorankikoto.hu). Apartments sleeping four from HUF6,500 (£19) per night, room only.
* Vital Hotel Nautius, Lake Velence region (00 36 22 570 115; hotelnautis.hu). Doubles from HUF26,000 (£76), including breakfast.
* Sarffy House, Dorgisce, Lake Balaton region (00 36 70 770 6945; sarffyhaz.hu). Doubles from €80, including breakfast.
* Tornacos Pension, Hegyko, Lake Ferto region (00 36 99 540 200; tornacos.hu). Doubles from HUF8,000 (£23), including breakfast.
Eating & drinking there
* Aranysarkany Restaurant, Szentendre, Danube Bend region (00 36 26 301 479; aranysarkany.hu).
* Raspi Restaurant, Lake Ferto region (00 36 99 355 146; raspi.hu).