The first in the Bond franchise is only really remarkable in light of its legacy.
James Bond is a legendary figure of pop culture. First envisioned by author Ian Flemming in 1953 in his novel, Casino Royale, the name has become synonymous with the film series, and the universally recognized catch phrase; "Bond... James Bond." Dr. No was made in 1962, and was the first in a long, long, line of films featuring James Bond, with several actors playing the role of the central character (and others) throughout the 50 year run.
The first film opens unexpectedly with three goofy looking old men crossing the street, to the tune of 'Three Blind Mice'. This strangeness abruptly gives way to thriller material when the three men break into a building and kill everyone inside. This prompts MI6, to send their best man. Here we meet James Bond, suave agent of British Intelligence. He's introduced to us in a nightclub, playing poker, because that's what gentlemen do. We see him briefed by his superior, 'M', and given a different pistol by 'Q' (a shadow of the scene in later films), and flown off to Jamaica to investigate.
In a 1962 interview, Flemming stated that Bond was originally conceived as an "extremely dull, uninteresting man to whom things happened." This is unfamiliar to Bond as made famous by the films, but rings more true in this film then (perhaps) any other. Later films, starting with the second, make Bond something of an invincible super-agent, but in Dr. No he is quite grounded, relatively passive, and a more mundane character.
Sean Connery is excellent at delivering smooth talk and dropping one liners, but in routine conversation he sounds a bit awkward - like he's clamouring to overact, but has to restrain himself, so every line is as forcibly dull as can be. This isn't a terrible problem, but being used to the slick pace and tight dialogue of newer films, it irked me a little. Likewise, action scenes in this movie have an old-timey awkwardness about them; humble beginnings for a franchise that would become known for its over-the-top action.
As for the espionage and the villain and his evil plan... I hardly followed. I can't recall the motive behind the murder that Bond is sent to investigate, I'm not sure how the woman was connected to Dr. No, and I have absolutely no idea what Dr. No was trying to accomplish (later, I read that he is planning to disrupt a rocket launch from his secret base. The motive? He's a terrorist, I guess).
The film didn't really seemed all that concerned about whether or not we were following the complexities of the plot; only that we were entertained. To that end they deliver the action beats, attempted murders, gratuitous bikinis, and bizarre evil lairs that are hallmarks on the series, but it all feels like it's still in development (with the exception of Honey Ryder, who seems to be the definitive 'Bond Girl'). I know people love this movie, but I found it slow and mediocre, and only really remarkable for its iconicism, and the fact that it spawned a pop-culture juggernaut.