Even in Britain's highest and wildest places it's not uncommon to come across a child being carried in a sling or special backpack. The sight is generally applauded – a fell-walker of the future inducted early in pleasures to come.

Hopefully, it will turn out that way. But the infant carried today might too soon be expected to scale the heights unaided. There's a danger that overambition, by parents who put their own walking itinerary first, will stifle rather than stimulate a young person's interest in walking.

At any rate, that's the view my wife and I took when encouraging our children to walk. Sometimes with reins, toddler outings might just have exceeded a mile, and even later on, four or five miles was our usual maximum. The major mountains were abandoned. Scafell lost out to Lakeland's midget Sale Fell. Clad in smooth turf, ideal for scampering kids, Sale Fell also offered gambolling lambs, an oak wood where woodpeckers might be heard, and the remains of an abandoned church.

The format must have worked, as our children, two daughters and a son – all now middle-aged – blossomed into keen walkers. Their enthusiasm is now being passed on to our grandchildren – again two girls and a boy. Like us, their parents aim not to press their kids to walk too far or in too-demanding terrain. They try to select walks that provide interest, variety and fun.

Stepping stones, fossils, hollow trees, drifts of dry leaves: these all qualify as fun. The leaves are to be kicked through or used as ammunition for a leaf fight; the trees for climbing into and peering out from. Our photo albums are punctuated with pictures of the kids in various hollow trees. There's delight if a return visit finds the specimen still there, waiting to be re-entered.

For kids, there's delight too if a walk can be paused for some treat. Miss no chance to call at a village shop (sweets), a wayside pub (lemonade) or, best of all, an unexpected ice-cream van. Happy family walking is essentially walking with interruptions. Leave route marching to the peak-baggers.

That said, even with the youngest children, a walk needs an objective. A hilltop monument might do. Or a tarn. But best is a rocky summit. In my main walking patch, the North York Moors, the mini-Matterhorn of Roseberry Topping fits the bill perfectly. A family can be up and down the hill within two hours and, after standing on the small platform of naked rock that forms the summit, a young walker might well feel like a conqueror of Everest.

Some of the easiest family walks are in formal country estates – such as Fountains Abbey, featured overleaf. The glorious Cistercian ruin, beautifully situated in the Skell valley near Ripon, has been the centrepiece of the estate's landscaped grounds, including extensive water gardens and a deer park studded with great trees, since the 18th century.

Any number of walks can be devised, linking features such as follies, a rock tunnel and the so-called Valley of the Seven Bridges – small arched structures that take the twisting path back and forth across the Skell. But are there seven? The walk offers a chance to count them. Even better is to bring a picnic, for which we choose a suntrap glade away from the main tourist circuit. There's a large, fallen sweet chestnut tree that we've often adopted as bench and table.

Any sandwich remnants are welcomed by wildfowl from the large ornamental lake nearby. Meanwhile, the deer park will have offered many sightings of its three species of deer: get the kids to discover which is which. In autumn we like to gather conkers and sweet chestnuts. The abbey ruins are a maze for children, who afterwards might have little energy left to expend on a recently-introduced adventure playground.

But perhaps there's little need to worry that some youngsters, introduced to vigorous walking too early, can be put off. Kids often survive sterner tests. After one family walk, we discovered that our grand-daughter Molly, then just four, had walked for three miles with her wellington boots on the wrong feet. Now a teenager, she still seems eager to walk – though it may have been in revenge that she, with sister Rosie and cousin Ned, recently set such a cracking pace for their now doddering grandparents on yet another ascent of Roseberry Topping ...