I’m zig-zagging slowly down a beginners’ slope trying to avoid the three-year-olds zipping across my path and under the outstretched arms of a giant Mr Man. My two daughters, age 12 and 10, glide along behind me. As we pull to a stop I breathe a sigh of relief. I didn’t fall over that time.

“High five,” says Nico, our instructor. My daughters slide into position beside me, nonchalantly touching gloves with him as they do so. I try to reach over to his hand without going bottoms up again. I miss.

It’s a humbling experience learning to ski with your children. My daughters have skied once before, while I’ve done a few weeks snowboarding over the years, but have never been on skis. I was thinking that probably evened things out. I was wrong.

We’ve come for a five-day trip to Katschberg, a high mountain pass in the Austrian Alps a 90-minute drive south-east from Salzburg. The resort is at an altitude of between 1,640 and 2,220 metres, so the snow conditions are generally excellent. If the weather is too warm – as it was when we were there in December – snow cannons line most of the 40-odd miles of slopes, so snow on the pistes is guaranteed. I have no excuses.

Our four-star Hotel Cristallo, part of the Falkensteiner group of hotels across the southern Alps, has its own ski school and ski hire on site, while the gentle “blue” slopes are right outside the door, so as soon as we’re kitted out we’re ready to go.

Right from the start, I’m like a duck on ice, slipping and sliding all over the place, while the children perform each exercise Nico gives us faultlessly. I end the first day bruised in more ways than one.

Fortunately the Hotel Cristallo is the perfect place to recover my strength and resolve. As soon as the lifts close, cake is served in the bar area – a sticky array of chocolate, almond and cream squares. Then before dinner there’s time for some recovery pampering. For adults, an hour in the sauna, with its vitamin infusions, steam room and ice pool, leaves you glowing.

For children, there is a spa, games rooms and two pools. We put on our matching white towelling robes and wander down together to sit on wooden loungers by the pool. The children enjoy swimming from indoors to outdoors through a glass wall that opens on request, lying under the stars in the steaming water until their heads gets too cold.

The hotel dinner also aids my recuperation. We’re given our own table in the restaurant which we’re to use each night, and at breakfast each morning. We don’t quite understand the protocol the first evening, walking in to find an outlandish buffet of cured meats, roast vegetables, salads, pasta dishes, cheeses, dips, dressings ... even caviar. Despite being vegetarians it is impossible not to pile our plates up high. After we’ve eaten, a waiter comes over to our table – is that an amused grin? – and asks us what we want for the main course.

Each morning dawns crisp and clear – the resort’s location on the southern side of the Alps means lots of sunshine on clear days – and my daughters are raring to go as soon as the lifts open. There are 16 lifts in all and rarely any queues to get on, while the runs are wide and fairly empty, so my girls happily bomb up and down, as I labour on slowly.

Every now and then, my wife, who is an experienced skier, stops by to check on our progress. There are plenty of more challenging runs in Katschberg to keep her happy, including a four-mile run from the top of Aineck, the peak opposite our hotel. On our last day, I take the chairlift up there simply to admire the view, which is particularly stunning at sunset with a cup of warm glühwein from the cafe. My wife and children ski back down, while I take the lift.

Everyone keeps telling us to visit Katschberg’s advent trail – which opens each December, this year from 29 November until 25 December (it is closed, however, on 24 December). My kids are not exactly sold on the idea, but their eyes light up when, to get there, we have to board a horse-drawn sleigh, which pulls us up to a remote hillside. It’s dark by the time we arrive and so we’re given a lantern each and proceed on foot. Along a two-mile snowy path are numerous wooden huts. Outside each one are people dishing out mulled juice and wine, while inside is a surprise – such as a storyteller, a weaver, or some children playing music. It’s a village-sized advent calendar.

Our favourite is a hut with four men singing Austrian folk songs. It is like stepping into another time, a hazy, idealised, Alpine past. We sit in the amber light of the hut, cradling our drinks, mesmerised by their voices.

Back on the slopes, eventually something clicks. Nico finds the explanation that makes sense to me – it involves looking over an imaginary fence and then ducking back down. It works. I’m turning, I’m skiing.

I tell my daughters I’m king of the slopes, the fastest skis in the west. They find it hilarious. “You just do snow plow the whole way,” they say, shaking their heads.

Of course, as I progress, so do they, and so I’m forever the slow one. But their jollity makes me happy, and I’m glad that together we’re all learning, out in the fresh air, our cheeks red and rosy. It’s certainly a lovely way to share time with your children.

• Prices at the Falkensteiner Hotel Cristallo from €966 for three nights full board including advent trail for two adults and two children under six, or €1,110 for three nights for two adults and two older children