SAY WHAT you want about Griff Rhys Jones – I find "Get this man off my telly!" is sufficient for my needs – but he doesn't shirk a challenge. New York is familiar even to people who've never been there. Most of us know a New York song, be it a lullaby of Broadway or gangsta rap best heard in a bouncing pimped ride. And Manhattan's cloudbusting skyline must be the laziest shot in all of traveloguery.

Incredibly, ITV thinks there's something new to say about the Big Apple, this despite the fact that when you Google "New York" and "famous quotes", the listings top 2,870,000. Rudyard Kipling, Henry Miller and, of course, Alistair Cooke have pronounced eloquently on the place – Robert Louis Stevenson, Jessica Mitford and Saul Bellow, too. Undaunted, though, our man seized the day.

For the first of Greatest Cities Of The World With Griff Rhys Jones, he travelled light. I'm guessing, but I would say a Berlitz guidebook, and that's all. Not the expanded version of four extra pages either. Oh, and he only read from the introduction: 8.2 million inhabitants, more than 170 languages spoken, baby born every four and a half minutes, blah-blah, yadda-yadda. Actually, I'm being unkind to Berlitz and their fanny-pack marvels of fact and tipping etiquette. This was New York as it's described in a dog-eared in-flight magazine stuck to a sickbag.

The programme blurb promised that Rhys Jones would get "under the skin" of New York by observing it over a single day, beginning in a Brooklyn diner. As is the way of these personality-led documentaries, the star helped out behind the counter. Then he joined the rubbish collection on the Upper West Side where he showed a surprising amount of concern for the fate of a discarded toilet bowl. We learned that this piss-pot would be transported to New Jersey, crushed and loaded onto a train to travel an "astonishing" 300 miles to a landfill site in West Virginia. Regular readers of this column will know what I think of Rhys Jones' presentational style, but I refuse to accept that at this moment I was the only viewer not wondering if there might be room in that crusher, and in that big hole in the ground, for Greatest Cities Of The World.

If you stuck with it, you will know that New Yorkers "love a good parade", that "the scars of 9/11 run deep" and that the city boasts 4,493 skyscrapers. For a moment, as he gawped upwards, Rhys Jones threatened to get philosophical. "What's up there? What's it all for?" he asked. But the feeling soon passed. By the end, he was back where he felt most comfortable: in a giant heating plant, enthusing about New York's plumbing – "The most impressive array of pipework I've ever encountered." So, you're wondering, if this programme was a New York song, would it be Frank Sinatra or Gerard Kenny? I can't believe you're even asking.

Greatest Cities Of The World ... (with Paris and London to come) has been scheduled against another travelling comedian and the most memorable image in the first instalment of Paul Merton In India was of a sadhu holy man winding his penis round a walking stick and a friend balancing on this makeshift perch, testing the disciple's devotion to the limit. The show's guide looked on in mild horror, but if you remembered last year's Paul Merton In China when he was served a meal of donkey penis, you might have reckoned this was just another funster, obsessed with pipework.

Merton, though, at least tries to come at travel reporting differently. "India – a country of a billion people… a landscape of stunning vistas… but I'm not going to show you any of that," he said in the introductory voiceover. Yes, it's a nuclear superpower – but as he pointed out it also worships snakes. As interested in the snakes, you guessed, he promised to avoid "the usual tourist trail".

There were no snakes in the opener but plenty of hooligan monkeys which inadvertently caused the death of New Delhi's deputy mayor while he was chasing them across rooftops. A tragedy, but behind Merton's sweaty brow you could almost see the cogs turning in his comedy brain – and when he witnessed bonkers Indian officialdom's attempts to curb the problem with a task force of trained apes, they went into overdrive.

Merton said he was raised into Catholicism but hadn't practised for years as he'd found it "too rigid". He thought Hinduism might suit him better, and although there was an obviously rigidity to that naked believer's walking stick, Merton entered into the spirit of a festival in Junagadh by smoking some dope. Presumably this would heighten the senses at the event's climax. "So let me get this right," he said, "I'm entering Shiva's wife's womb to see his penis impregnate her?" Yes, he decided, Hinduism was "fun". As was this programme.

Tommy Cooper stuck with the brilliantly rubbish jokes right through his career and died on stage. He might wonder about the staying power of the modern gagman; Griff Rhys Jones doesn't do comedy any more, at least not intentionally, and Paul Merton restricts himself to going off on one, hilariously, on Have I Got News For You. Steve Coogan is younger than them, also funnier, and so shouldn't have to resort to travelogues just yet. But he does have a habit of following a great show (most recently Saxondale) with a much lesser work, and Sunshine got off to an overcast start.

Coogan didn't write this three-parter; it's the work of the Early Doors team of Craig Cash and Phil Mealey. They also act in it, playing the binmen sidekicks of Bob "Bing" Crosby (Coogan), a charming chancer with a heavy gambling addiction. Unfortunately, Sunshine is sentimental when you don't want it to be, not tough enough where it needs to be (such as when our hapless hero gets lured into an illegal betting ring), and only funny when Bing's dad played by Bernard Hill regales his grandson with his war exploits – and the Chinese Burn that did for Hitler.