It’s been called the new space race – a multi-billion dollar battle to offer intergalactic tourism to anyone who can afford it. Sir Richard Branson, Elon Musk and Amazon founder Jeff Bezos have all staked their sizeable reputations on making historic flights into the unknown.

The trio claim their individual space projects will see man boldly go skyward by the end of 2018. Branson hopes to be first with his Virgin Galactic programme, based in New Mexico. Tom Hanks and Brad Pitt are already signed up for the adventure of a lifetime.

Musk’s SpaceX team is working with NASA to launch from Kennedy Space Centre in Florida. The mission is to send two as yet unnamed civilians around the Moon. The man who founded PayPal and electric car company Tesla wants to make space travel as easy as hopping on an aeroplane.

Meanwhile, Bezos claims that he will be first with Blue Origin. It plans to start commercial, suborbital flights early next year (2018) from a base in Texas. Named the world’s richest man in July, Bezos started dreaming of space as a five-year-old, watching Neil Armstrong walk on the Moon.

If that’s not enough to get would-be space cadets drooling, the Apollo 17 lunar rover abandoned on the Moon in 1972 is about to get its first visitor. A private lunar mission is due to arrive near the Taurus-Littrow next year, equipped with two vehicles built by Audi.

The lunar quattros will travel to the Moon on board one of SpaceX’s Falcon 9 rockets. Designed in conjunction with German company PT Scientists, the all-wheel drive rovers will explore the area before sending HD quality imagery of the Apollo 17 buggy back to Earth.

Audi spokesman Michael Schoffmann explained: “It is a quattro, like many of our road cars. It also has an e-tron battery, like the ones that we use on Earth. But we are breaking new ground with the rover and can learn how automotive components behave in such extreme conditions.”

Space tourists will look back at our planet and see just how fragile it is. Obviously there is nothing like it on Earth

“We choose to go to the Moon”, exclaimed President Kennedy when he kick-started the original space race back in 1962. That’s all very well if you’re a NASA astronaut or a tycoon with deep pockets – but what about the rest of us who can’t afford such heavenly prices?

To explore space travel from terra firma, I’ve organised a 2,000-mile road trip across America that includes some of the best space-related tourist venues in the country. It begins at Kennedy Space Centre in Florida and will cross seven states to Flagstaff, Arizona – where astronauts trained to drive the original lunar buggy back in the 1960s.

The concierge at the Ritz-Carlton Orlando is doing his best to explain how it feels to watch a rocket launch. He beats his chest with two fists and becomes more and more animated. “You can really feel your heart pumping – the thrust from the engines vibrates right through the landscape. There’s a flash of fire, the ground seems to move and then the flame soars into the sky.”

A rocket launch is visible from almost anywhere on the Florida peninsula. Some take their cars to New Smyrna Beach and soak up the atmosphere with a barbecue, others make their way to Kennedy Space Centre Visitor Complex, the closest and busiest viewing spot.

I timed my space race trip to coincide with a rocket launch but the mission has been delayed. Instead, NASA astronaut Sam Durrance tells me what lift-off feels like from the inside. “It’s like hanging on to a runaway train at 17,000mph. You can’t stop it – there’s no getting off.”

Lunch With An Astronaut is one of the most popular activities available at Kennedy Space Centre. Durrance is an American scientist who spent 25 days in space on two missions to the International Space Station. He is still visibly moved by the experience during our conversation.

“Space tourists will look back at our planet and see just how fragile it is. Obviously there is nothing like it on Earth. I’m too old to ever go back now but what I experienced up there will live with me forever.”

Durrance has degrees in physics and astro-geophysics but what are the most common questions people ask at the dinner table? “They usually want to know how we go to the toilet in space – and what the food tastes like. Neither answer is very edifying.”

From Orlando it’s a nine-hour drive to the John C Stennis Space Centre – a rocket base on the Mississippi-Louisiana border that was used as a test centre for the Apollo Program.

Space Centre Houston has an overwhelming number of displays, all dwarfed by the 747 aircraft parked outside and the Shuttle replica on top of it

I’m at the wheel of a hired Lamborghini Huracán, the closest I could get to a road-going rocket ship. It’s based on the same platform as the Audi R8, features a matching V10 engine and is also equipped with quattro four-wheel drive. A pearlescent Nasa white, it’s as close as I’ll get to driving a spaceship – and given its mighty acceleration (0-60mph in 3.4 seconds) and prodigious handling, I don’t think the experience is a million miles off.

Nearby New Orleans is a good base from where to visit Stennis. It’s also home to the Michoud Assembly Facility, where the first stage engines of the Apollo rockets were built. Later, the enormous external fuel tanks for the Shuttle were constructed here, too.

The Infinity Science Centre is the official visitor complex at Stennis and offers an educational approach, compared to the theme park razzmatazz of Kennedy. A guide confides that the best time to visit is when a rocket test is taking place, although it’s pot-luck choosing because there is no official test schedule.

When I arrive in Houston five hours later, the Buffalo Bayou river has flooded. I’m staying at The Sam hotel in the city centre, unaware that two weeks later the entire area will be devastated by Hurricane Harvey. The historic hotel survived and even managed to stay partly open through much of the disaster.

Johnson Space Centre on the outskirts of the city is where NASA’s mission control directed flights from the launch pad at Kennedy Space Centre. Its visitor complex – Space Centre Houston – has an overwhelming number of displays, all dwarfed by the 747 that’s aircraft parked outside and the Shuttle replica loaded on top of it.

Driving west again through San Antonio and onwards along the Mexican border, even the scenery in this part of Texas seems a little space age. The huge skylines and lunar-style desert have turned the sandscape into an otherworldly sort of place, where little seems to survive the extreme heat.

That evening, sat in the V6 coffee bar at the laid-back Gage hotel, in Marathon, I’m told about strange lights that sometimes appear over the desert in the neighbouring town of Marfa. Everybody seems to have an alien story to tell here. And as if to prove it, just across the border in New Mexico is the International UFO Museum.

Seeing the original boot marks in the dust of the Moon could be the highlight of future space tourism

The next day, en route to El Paso I visit the McDonald Observatory, near Fort Davis. In the wilderness of Texas, the dome-like structures look like they could have landed from another planet. The weekly Star Parties aren’t a chance to enjoy cocktails with Hollywood elite but an educational tour of space with a well-briefed boffin.

The final 800-mile drive west on the I-10 takes two days. Temperatures top 40°C and the road often disappears in an illusion act caused by the heat haze. My final destination is Arizona – and for good reason.

The rocky landscape that’s outside my window at the luxurious Phoenician hotel near Phoenix is a clue. Located just up the road from the Waldorf Astoria resort at Flagstaff is the spot where an old volcano was transformed into an astronaut training ground in 1967 to trial the original lunar rover.

Thousands of tons of explosives were used to recreate the craters of the Sea of Tranquillity, the site where Armstrong and Aldrin would make their historic landing two years later. NASA used detailed satellite photographs of the Moon to get the terrain just right – although as I stand next to the area today, a relentless wind has reduced the surface to flat desert again.

The footprints of astronauts who trained here during the Apollo era have long disappeared. Only a handful of the 12 men who walked on the surface are still alive today – Armstrong himself died in 2012.

Seeing their original boot marks in the dust of the Moon could be the highlight of future space tourism. And think of the Air Miles you could clock up on a 478,000-mile round trip.

Hayes & Jarvis offers a 13-night USA road trip from Orlando to Phoenix on a room-only basis from £2,195 per person. Includes ‘standard’ car hire and return flights from Gatwick with British Airways. Hire a Lamborghini from from £1,160 a day.