It is 24 hours before the over-60s can use their new freedom passes and the sense of anticipation is infectious. Almost. “We have to travel light,” says my dad, and I can tell that he's excited. “There's nowhere on buses to store luggage safely, you have to keep it with you.” Fishing in his wardrobe, he extracts an ancient backpack no bigger than Dick Whittington's handkerchief, tosses in a toothbrush, a pair of underpants, his heart pills and a 1994 guide to the North York Moors, and we're off, him striding from his home in the West Yorkshire rhubarb triangle to the bus stop, me scuttling behind. “There's the first interesting historical feature,” he says, gesturing at a mound in a field. “An early 19th-century spoil heap.”

I am transported back to my childhood when my brother and I were carted around in the rear of a Vauxhall Viva by our dad, then a history teacher, in search of ancient monuments. Few Bronze Age earthworks escaped us. This week's introduction of the English National Concessionary Pass scheme, enabling the over-60s to cavort around the country on buses without paying a penny, is my chance for payback. Dad, very nearly 79, has agreed to join me on a two-day 140-mile round trip in which we will possibly not spot many ancient monuments at all, apart, that is, from fellow passengers.

At Leeds bus station, we are yelled at by a man with a megaphone who insists that we stay behind automatic doors. Above each one a scary sign says “DANGER do not attempt to board moving buses”. I catch a whiff of vomit. All this is perhaps why there are a lot of people here who look over 60, even though they're probably not.

The 10.15am Yorkshire Coastliner to Thornton le Dale is on time and we have the top deck to ourselves. Immediately we realise that the perk of bus travel is that you can see all the things you miss in a car. Santa and his sleigh are still on the top of a pub, a beautiful stone angel hangs over another pub door, we glimpse a fine collection of garden gnomes and the interiors of a good many bedrooms.

You also realise that if a sign says left to York, you will go right through a housing estate or village. There are no historical monuments here so by 10.50 we have discussed violence among the young, the licensing of aggressive behaviour in Nazi Germany, The Simpsons and the psychology of Pinocchio. I am not sure I have the stamina for this trip.

In York we pick up a group of youths, which provides entertainment of a kind. Swagger is fat, wears an Arsenal shirt and moves around between the two decks. Glum, a rat-faced boy, is playing tinny music on a mobile phone and has turned up the volume. A list of rules on the bus ceiling makes it clear that this is unacceptable, but we've passed Castle Howard and gone through Malton before I dare to ask him to turn it down. “Stop that crap music, mate,” says Swagger to Glum, hitting him over the head with a plastic water bottle. They get off at Kirby Misperton. If we were on a plane we could be in Tuscany by now.

We alight at Pickering, where there is a handy public loo, and buy cheese rolls in a greasy spoon where an elderly man says he won't be using his freedom pass because he'll get lost. “Wouldn't be able to find the bus stop, me.” We catch the 13.12 to Helmsley, which we reach at 14.00. We have spent four hours and 40 minutes travelling about 75 miles, meaning that we have averaged 16 miles an hour. There are times on this trip when it strikes me that local bus travel is like going to a carvery where you can eat as much as you want for £6.99, and realise after the first plateful that you've had enough. “Let's go to the castle,” says Dad.

The castle houses the tourist office. Is it geared up for an influx of over-60 visitors, we ask. Not really, they say, but they find us a B&B for £37.50 a head. Extra costs are the limitation of the scheme. Yes, you can go anywhere without having to pay, but unless you really want to sit on a bus all day to get home for bedtime, you may pick up accommodation costs. “Can you go to London?” asks a lady in her mid-80s. Yes, I explain, but it will take a few days. She thinks she'll just use her pass when she stays with relatives, and that's the conclusion reached by most of the people we meet. “Not a castle with much military use,” says Dad, as he scrambles up and down the dry moats. “The castle held out to the Roundheads for three months in 1644.” I can tell that he's happy, not least because he likes the idea of not spending any money. “I'll be able to go to York and not pay £12 for a day's parking,” he says with relish.

The following morning, the day of the concession launch, Helmsley's market square is deserted at 9.30am when the X31 to York arrives. According to the propaganda put about by bus companies, there were supposed to be hordes of oldies battling for seats on every bus. We feel deflated and our mood is not helped by the driver, who says that because Dad's new pass hasn't arrived, he'll have to pay. But the driver makes a phone call and refunds the money, which produces wreathes of smiles from the over-60s on the bus. Though, to be fair, they were smiling already. “We like something for nothing,” says a lady in her 70s. “There's only one drawback - not enough bus shelters. But I think this will get a few off the road. I can think of plenty who shouldn't be driving.”

We meet John Thompson, a retired computer programmer from Kirkbymoorside who turned 60 six weeks ago and who has spent most of that time on buses. He's a walker, too, but clearly the buses offer him challenges that he feels powerless to resist. There's not much that he doesn't know about the local timetables but today is his first chance to explore farther afield and his bright blue eyes are glittering. When bus riding is declared a sport, John will be an avid competitor, I am sure. Connections permitting, he's going to Doncaster, where he will spend an hour before heading home for tea at 6.30pm. “It's gone very wrong on two days,” he says, adding something about the X46 to Hull. But he's had some super trips, seen frosty scenes that looked like fairyland. “Today I'm stitching three buses in series, which is a difficult thing to do. This is a new lease of life.”

Our route is through pretty villages - Ampleforth, Byland with its spectacular abbey ruins, Coxwold with the octagonal church tower - all set against the fine backdrop of the Howardian hills. Dad is in raptures. He drove along this road recently but is seeing so much more. His fellow over-60s are chattering away, mostly to people they don't know. “Of all the things they could spend money on - aren't we lucky?” beams one lady. “That was a great pleasure,” says Dad. It is the men who are most openly enthusiastic, perhaps because in this generation men are more confident than women about talking to strangers. When we reach York, John Thompson bounds off to crush a seven minute walk into five and, he hopes, make his connection, and Gideon Kidd, a retired chef of 85, dashes to the Guildhall to collect his pass in person. There he is one of several hundred people queueing for the 28,000 passes that need to be distributed in York - it is finally handed to him after an hour and 40 minutes. “I won't go far,” he says, but he can't stop grinning.

Dad has made some fresh historical discoveries by following a quirky guide to York and his mood is dampened only by the jobsworth tones of the next driver who refuses to accept his old pass and insists that he pay. At Leeds, Dad fills in a complaint form, not to request a refund, but to register the driver's rudeness. He feels vindicated, though I can't say I do when the friendly woman at customer services says, “Oh, you must be Mrs Wark, your pass is in the post”.

After two days and £23.10 worth of bus travel, I am clearly morphing into a relic, which shows, if nothing else, that I am my father's daughter.

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