The pretty Yorkshire Dales town of Hawes looks as if it has come down with a nasty case of measles. The white facade of JR Hopper estate agents (est. 1886) is spattered with red spots. The post office is festooned in polka-dotted bunting. At the White Hart Inn, the landlord is attempting to perfect a cocktail that combines a White Russian spiked with drops of Campari. "It's certainly got a kick to it," he winces, but his face falls as the drink turns from spotty to uniformly pink. "It needs something a bit more syrupy than Campari."

It's not measles, of course, but something closer to Tour de France fever that Hawes is experiencing. On Saturday, 5 July, the world's most famous cycling race is coming to Yorkshire for two days and the town is set to play a pivotal role. It is nestled at the base of the steepest climb on stage one – hence it is favouring polka dots in honour of the shirt worn by the leader of the King of the Mountains classification, over the traditional yellow jersey of the overall lead rider – and is sure to be a popular congregation point for some of the million spectators expected to line the roads on that day alone.

It was announced in December 2012 that Yorkshire had been selected to host the Tour's grand départ. This was a shock: the county faced competition from Barcelona, Berlin, Florence and Scotland, which was backed by the government and British Cycling. Whether or not you were interested in the sport, the news had economic repercussions. For an outlay of £27m – to host a pair of stages in Yorkshire and one from Cambridge to London – it is projected that local businesses will make more than £100m. Not a bad return for three days' work. If this sounds too good to be true, some think it might just be so. A Yorkshire Post poll asked readers, "Will the Tour de France be good for Yorkshire?" More than 1,300 voted, with 71% saying yes and 29% no. I came to Yorkshire to cycle the 119-mile opening stage from Leeds to Harrogate, but also to find out how people feel about the race and why suspicion still lingers.

Stage one starts, as do I, at Harewood House, an ancestral pile on the outskirts of Leeds noted for its Thomas Chippendale furniture, Capability Brown gardens and its incongruous colony of penguins. The house is occupied by David Lascelles, eighth Earl of Harewood, and it is from this bucolic vantage point that Princes William and Harry and the Duchess of Cambridge are expected to watch a race that will cover 2,272 miles in all and end three weeks later in Paris.

My companion for the first phase of the ride is a genial, 40-year-old Geordie called Chris Clyburn, who has lived in Leeds for half his life. Clyburn is a "tour maker", an initiative the race copied from the volunteer games makers who were such a tirelessly upbeat presence at London 2012. When the call went out, more than 30,000 people responded and this number has been whittled down by availability, suitability and criminal-record checks to a formidable army of 10,000 in lurid green shirts, working across the three days the race is in the UK.

As we roll north from Harewood House, along the River Wharfe towards Otley and then Ilkley, I ask Clyburn why he signed up. He had been a cyclist as a teenager, he said, and lost interest as he became a chef and then tried for a career in music. But when there was a party in January 2013 to celebrate Yorkshire winning the bid, he headed down "with a few other idiots" to Leeds town hall in the driving snow. The only explanation he can think of is civic pride.

"People ask me, 'Are you really just going to stand by the road, waving a flag and not getting paid for it?'" says Clyburn. "But honestly, I'd have paid to do it. If I can make one person's day better – because I show a child where the toilets are or whatever it is – I'll be happy, because I'll be part of the legacy of cycling in Yorkshire."

From Ilkley, the route heads into the Dales through Skipton, the best place to live in Britain, according to a recent survey. Clyburn says many of the Tour makers are former games makers keen to recreate the camaraderie from that event. Their four-hour briefing sessions covered everything from identifying a terrorist threat to protecting the centuries-old stone walls.

At Kettlewell, renowned for its scarecrow festival, a straw man is dressed in a maillot jaune (yellow jersey). As you leave the village, the road narrows and for the first time the Tour riders might breathe a little harder as they climb Côte de Cray. (All of the ascents are temporarily infused with a Gallic grandeur, including, brilliantly, Côte de Buttertubs.)

Côte de Cray will be nothing for the professionals, but it's steep enough to leave Clyburn and me wheezing. "Are you allowed swearing in the Observer?" he asks as we reach the top.

Tim Lewis, with Dan Atkinson and Pete Flynn, climbs Buttertubs on the route of stage 1 of this year's Tour de France. Photograph: Gary Calton for the Observer

Clyburn heads home and after a night in Hawes, about halfway through stage one, I'm joined by two members of the Cappuccino Cycling Club in Harrogate: Dan Atkinson and Pete Flynn. Both cheerfully admit to being Mamils (middle-aged men in Lycra) who are riding bikes that are "probably £2,000 better than we actually need". They are emblematic of a road-cycling boom in the UK that now sees Harrogate have three vibrant clubs with about 600 members – including a good proportion of women – in total.

It is a stunning day and, as we ride through the Dales, Yorkshire's self-awarded nickname, God's Own County, starts to make a lot of sense. "The route the organisers have picked through Yorkshire is really outstanding," says Flynn. "Day one is all about a whole series of stunning dales, winding through to Ripon and Harrogate, the old spa town, representing the leisure side. On the second day, the race goes through a lot of the old mill towns. As you climb out of Keighley, the economic history of the region will be very obvious to the riders. It really shows off Yorkshire at its best."

The pride of Yorkshire is well-known, but as we ride over Buttertubs pass – so named because farmers would stop here on the way to market and lower their produce to cool in the fluted limestone potholes – it is clear that many in the county feel this is a long-awaited opportunity to prove it has moved on from the Hovis adverts. The Olympics was London's party; the Tour de France gives Yorkshire the opportunity to hog the limelight in front of a global television audience.

"There will be a new understanding of Yorkshire," predicts Atkinson. "The London-centric view of the world is like, 'Ey up, it's grim up north.' It's a little bit comedic and a lot of the TV shows play on those stereotypes. But there's something special and natural about it and hopefully the Tour is going to show that."

Everywhere we stopped, some people were enthusiastic about the Tour's arrival and a few couldn't care less. In the village of Gunnerside, a woman was turning her garage into a bunkroom and having 42 people to stay over the weekend. In Hawes, a B&B owner complained that the local authorities had dug up the road outside his house for two weeks and noted with disappointment that Bradley Wiggins – one of the champions of Yorkshire's bid – wasn't even going to be selected by Team Sky. Farmers are annoyed that they will in effect be cut off by the road closures on the day and delayed during a crucial period of the hay-making season. Drivers moan that they have to share the pristine new stretches of tarmac with "road-tax-dodging hippies" (despite the fact that "road tax" was abolished in 1937).

All major sporting events come with similar gripes and the hope is that they will be forgotten once the action starts. Certainly, as we rode into Harrogate, weary from a long spell in the saddle, it was impossible not to be uplifted by the miniature hand-knitted jerseys that are strung around the town to welcome the Tour riders. The borough council launched the Tour de Bunting project last November asking for volunteers to knit 2,300 jerseys. They ended up with 23,453, with contributions coming in from the Arctic circle to New Zealand.

I was reminded of an advertising hoarding for the Yorkshire Post I'd seen 100 miles earlier in Otley. It read: "The French believe they have the toughest summits. Our view: it's all downhill after Yorkshire."