The films can be divided in to a couple of sub-sections, like
Pre-talkies: Ingeborg Holm (1913) - The Girl in Tails (Flickan i frack 1926)
The classical era: One Night (En natt 1931) - Wild Strawberries (Smultronstället 1957)
Experiments and new waves: Raven's End (Kvarteret Korpen 1963) - A Swedish Love Story (En kärlekshistoria 1969)
Modes of realism: The Emigrants (Utvandrarna 1971) - Children's Island (Barnens ö 1980)
The present day: Four Shades of Brown (Fyra nyanser av brunt 2004) - Sebbe (2010)
Seeing at least one, anyone, from each sub-section would then be a quick crash course on Swedish cinema history.
But to get more depth, here are some suggestions and the reasons for seeing them (I have deliberately left out Bergman's films, not because I don't like them, but because Bergman is well-known as it is):
1) Ingeborg Holm is one of the greatest silent films ever made. It has restrained acting, vivid use of space and depth-of-field and a powerful moving story. It's also an angry film, about the treatment of the poor by the state, which led to a big debate and eventually to some laws being changed. It's also an early film by one of cinema's most important and influential filmmakers, Victor Sjöström.
2) Thomas Graal's Best Film (Thomas Graals bästa film 1917) is a cooperation between Sjöström, Mauritz Stiller and Henrik Jaenzon, three of the six creative forces that made early Swedish cinema such an impressive period (the other three are Henrik's more famous brother Julius Jaenzon, Georg af Klercker and Gustaf Molander), so that should be reason enough to watch it. But it's also a fun, modernist film about films, dreams and make-believe. It stands together with Buster Keaton's Sherlock Jr. (1924) as an early, playful, investigation into what cinema is and can do.
3) One Night is a formal experiment by Gustaf Molander and his constant cinematographer Åke Dalhqvist. It might be called the odd film out, both in Molander's career and Swedish cinema of the time, but as an experiment it's bold and interesting. The inspiration comes from the Russians, Eisenstein and Pudovkin.
4) Karl-Fredrik Reigns (Karl-Fredrik regerar 1934) is less interesting as a film than as a historic document about the Social Democratic mood in Sweden at the time, with the building of the welfare state and what the Prime minister Per-Albin Hansson called "folkhemmet" (literally "the people's home" but perhaps more accurately "the people's society"). A number of films were made to sell the story about the "folkhem", more often than not directed by Gustaf Edgren. But there's also good acting so it's enjoyable as a film too.
5) A Woman's Face (En kvinnas ansikte 1938) is more typical Molander than One Night, and one of many films he made with Ingrid Bergman. And Åke Dahlqvist is the cinematographer. The look of Molander/Dahlqvist's films is usually sombre, with expressive use of heavy shadows and pools of darkness, and En kvinnas ansikte is not an exception. It's also typical for Molander to be about a woman's face, a woman's psyche. Before Bergman Molander was the preeminent "women's director" (whatever that is) in Swedish cinema.
6) Changing Trains (Ombyte av tåg 1943) is the first Ekman film in this retrospective, and it's also the first of Ekman's great films, the first of his theatre films. It's also a deeply autobiographic film, with Ekman playing more or less a version of himself, and Sonja Wigert playing a character that's a thinly disguised portrait of Ekman's long lost love Tutta Rolf. It's also so happens that it's one of the most moving of Swedish films. That Ekman was influenced by French poetic realism can also be seen.
7) The Girl With Hyacinths (Flicka och hyacinter 1950) was Ekman's own favourite among his films, and it was Bergman's favourite too. In fact, it's probably most people's Ekman favourite. Ekman was an actor's director, but he was also very cinematic, perhaps the most naturally gifted visual storyteller Sweden had ever had, and these two qualities are both clearly visible in The Girl With Hyacinths. The opening sequence, from the scene at the party to the arrival of the boy on a tricycle, could be thought in film schools. Like Changing Trains, it's also biographical, but in a more complex way, and can be seen as a comment on Ekman's and Eva Henning's relationship, with Ekman letting the various men The Girl encounters represent different sides of him. It's also a landmark of sorts in Swedish queer cinema.
8) Miss Julie (Fröken Julie 1951) was one of the films that showed the world that Swedish cinema again was a force to be reckoned with, after winning the Palmé d'or in Cannes (together with Miracle in Milan (Miracolo a Milano 1951). Miss Julie is Alf Sjöberg's version of Strindberg's play, and it's a tour-de-force of both acting and mise-en-scène, where the past and the present are shown in the same frame but on different levels in depth. Like Bergman, Sjöberg moved back and forth between the screen and the stage. He was one of the most inventive and artistic of filmmakers, and this arguably is his masterpiece.
9) Kisses & Hugs (Puss och kram 1967) is a film about men, women, and their relationships. It's directed by Jonas Cornell (who has said he was inspired by Hawks's Hatari! (1962) so naturally I like it) and it's part of the new wave of Swedish cinema, together with the likes of Bo Widerberg, Vilgot Sjöman and Jan Troell. This is Cornell's first film, and after a few inventive films he shifted to TV and more commercial productions.
10) A Swedish Love Story (En kärlekshistoria 1969) is Roy Andersson's first feature and I believe it is his best. It's unmistakingly Andersson, but it isn't as stiff and hasn't the annoyingly stylised theatricality of his later work (and no white faces either). What it has though is both a tender love story between two teenagers and a sad portrait of their parents, or rather the adult world. Adulthood is here a different country, and it has so much agony and despair it's painful to watch.
11) Man on the Roof (Mannen på taket 1976) is a police thriller, a Swedish version of French Connection (1971) or Madigan (1967). It's deceptively slow, after an initial violent murder, but works its way methodical and with a great attention to detail to the sudden outburst of violence in the second half, when the usually so peaceful Stockholm becomes a city under siege, terrorized by a heavily armed sniper. It's worth remembering though that Sweden, and Stockholm in particular, in the 1970s had seen several terrorist attacks, hijackings and spectacular bank robberies which had somewhat shattered the illusion of being at peace. Since the location shooting and realism is so striking, it's to this day difficult to walk the streets around Odenplan without looking up towards the rooftops.
12) The Girl (Flickan 2009) was to my eyes last year's best Swedish film. A haunting and ethereal story about a little girl left alone in a house during a few summer weeks. It moves back between dream and reality, and paints a not particularly flattering portrait of self absorbed adults. But unlike A Swedish Love Story it also has the courage to show that children can be equally cruel and corrupted. But perhaps the best thing about it is its cinematography (by Hoyte van Hoytema, the most celebrated cinematographer in Sweden today), which is primarily what gives it its haunting and ethereal quality.
13) The King of Ping Pong (Ping-pongkingen 2008) was short filmmaker Jens Jonsson's long anticipated first feature film. As was only to be expected it is beautifully shot, in several shades of white. But after a while that can become monotonous, and the story isn't particularly original or involving. So the main reason I included it here is that unlike any of the other films, it's set in the far north of Sweden, with endless vistas of frozen lakes, snow covered mountains and wide white plains.
There are a number of important female directors in Sweden, but only two has made it to this retrospective, Karin Swanström and Mai Zetterling. I haven't mentioned their films here because I haven't seen Swanström's The Girl in Tails and I don't particularly like Mai Zetterling's films (mostly because I feel that Zetterling is spending too much time and effort on editing and fancy camera work in a way that doesn't enhance the films but rather gets in the way). But she's still an important (feminist) filmmaker and deserves to be seen and discussed.
Also, the films I've listed aren't the only good ones, many of the others are equally great, but I had to draw the line somewhere. But if you have the time, don't miss out the third of the Ekman films being shown, The Banquet (Banketten 1948). It's possibly Ekman's most cruel and shocking film, shot like a noir with spider webs of shadows engulfing the sadomasochist couple in the centre. But good as it is, it's not as important as Changing Trains and The Girl With Hyacinths.