I'm sitting here thinking, "trains planes and automobiles", is that really an "old classics to watch"? I was 22 and in my first job after university when that film came out.
"...and if anyone has any recommendations..."
Well, for starters, I would pretty much avoid any film from the 1980s.
Well except, maybe, for the 1983 film "WarGames" where a very young Matthew Broderick played a nerdy teenage computer geek.
Or maybe "Beverley Hills Cop" (1984) a great action/comedy with Eddie Murphy (so many people had "Axel F" as a ringtone back then).
Speaking of Eddie Murphy, another film I would recommend is "Trading Places" (1983) with Eddie Murphy, Dan Aykroyd and Jamie Lee Curtis. A really great film.
And of course, speaking of Dan Aykroyd, you have to say, the original "Ghostbusters" (1984) is a really great film for a teenager to watch.
But I do remember that my younger brother said that he always also really liked a very overlooked film (that also starred Bill Murray and Harold Ramis) called "Stripes" (1981) where they joined the US army.
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Having said that though, here are some suggestions for some older films for a 13 year old boy.
I will start off with saying, just about any "Carry On..." film he will probably find funny (it will also give you an opportunity to talk afterwards about differing attitudes towards the roles of women).
I would likely start off with Carry On Spying (1964), Carry on Cowboy (1965), Carry On Up the Khyber (1968) and Carry On Cleo (1964). Some of these are currently on ITVX.
But, if you aren't happy with that (although he will likely enjoy them), how about:
"Robin Hood: Prince of Thieves" with Kevin Costner (currently on Netflix). It's even newer than the other film (1991) but is an enjoyable historical romp.
https://www.netflix.com/watch/60028308
Even though Kevin Costner was the lead (with Morgan Freeman as his sidekick), I absolutely loved Alan Rickman as the evil Sheriff of Nottingham. He had some really memorable lines like "Call off Christmas":
“That's it then. Cancel the kitchen scraps for lepers and orphans, no more merciful beheadings, and call off Christmas.”
In the original theatrical release of the film there was one very small part of a scene, that lasted maybe three or four seconds, that absolutely shocked me. But whenever I have seen the film on TV or streaming, that part of the scene (or often the entire scene) has always been removed.
Going back a bit further, how about Grease? The 1978 musical with John Travolta and Olivia Newton-John. I first saw this when I was 13 and my brother (who was 12 at the time) also really enjoyed it. It's currently on Amazon Prime.
You might also want to think about some "westerns"
For example, "Butch Cassidy and the Sundance Kid" (1969) with Paul Newman and Robert Redford, the film won four Oscars. A lot of action and also singing "Raindrops Keep Fallin' on My Head". This is a trailer:
Incidentally, also think about "The Sting" (1973) with the same actors. That won seven Oscars.
Or also "The Magnificent Seven" (1960).
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Also think about some of the early James Bond films. Anything from the first three films "Dr No" (1962), "From Russia With Love" (1963) and "Goldfinger" (1964)
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There are also some fun little comedies like "Those Magnificent Men in Their Flying Machines" (1965) about a fictitious air race race form London to Paris in 1910. It has Terry Thomas as the classic evil villain who eventually gets his due rewards. It was nominated for an Oscar for Best Original Screenplay in that year.
Then there was a follow up to that in the same vein called "Monte Carlo or Bust!" (1969)
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But, for me, "old classics" are slightly older than that.
Trying to think about what my brother might have liked at that age I think of the old Ealing comedies.
For example, the 1959 film "The Mouse that Roared" with Peter Sellers.
Or his later, much darker, film "Dr Strangelove" (1964) set during the Cold War.
Or how about "Kind Hearts and Coronets" (1949) a comedy/drama where Alec Guinness (Obi Wan Kenobi in Star Wars) plays several different characters that get killed off so that somebody can inherit a title.
An actress who appeared in that film, Joan Greenwood, also appeared in the film "Whisky Galore!" (1949) which is a really amusing film about a ship full of whisky being wrecked on a Scottish island during the war when there was rationing.
Or, "The Titfield Thunderbolt" (1953) a comedy which involved trying to keep a railway line running.
Or, "Passport to Pimlico" (1949) where the residents of Pimlico found out that they could declare independence from the UK and so not be subject to rationing anymore.
Or there are any number of films by the Boulting brothers. For example "Private's Progress" (1956) and "I'm All Right Jack" (1959) that starred Ian Carmichael, Richard Attenborough and Terry Thomas. Amusing films; the first is about life in the army and the second is about then contemporary industrial relations. Peter Sellers won the BAFTA for Best British Actor in a Leading Role for that film in 1959.
Incidentally, Jack Lemmon won the BAFTA for best Foreign Actor in that year for "Some Like It Hot" which is also another really good film that I think he might enjoy.
Talking about Ian Carmichael, another film worth watching is "School for Scoundrels" (1960) a comedy about a man who takes revenge against those who have taken advantage of him.
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Then another couple of suggestions. "Lawrence of Arabia" (1962) a really stunning film starring Peter O'Toole (along with Alec Guinness and a whole load of other famous actors) set during World War One.
Or a couple of films about thieves:
"The Pink Panther" (1963) the original film in the series with David Niven and Peter Sellers.
"To Catch A Thief" (1955) directed by Alfred Hitchcock and starring Cary Grant and Grace Kelly. Cary Grant is a retired burglar in Monaco who has an imposter copying his former method of operation.
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And then, just finally, something totally off the wall. A film I remember with pleasure that seemed to be shown on BBC every Christmas when I was young "Born Free" (1966) based on the true story of a couple in Kenya caring for an orphaned lion cub called Elsa.
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ps I'll just end all this by saying that my brother saw the film "Animal House" in the cinema at the age of 13 (he sneaked in under age, it was a "14" film) when it was first released in the UK. He enjoyed it very much at the time but says that on watching it now as an adult a few scenes did make him cringe and he would not want his own son to go and watch it at that age without him talking to him about it afterwards.