There's actually a fair amount of continuity in the early Bond films. I'd say all the Connery films and On Her Majesty's Secret Service are definitely part of the same continuity (despite the confusion in OHMSS created by Blofeld apparently not recognizing Bond). I'd also put Licence to Kill in this same continuity, as Bond's mission of vengeance against the killer of Felix's wife seems to be a vicarious mission of revenge for his wife.
Beyond that, yeah, they're pretty much standalone films. Casino Royale and Quantum of Solace are the first Bond films since the sixties that actively try to be a part of the same continuity.
I also disagree with the sentiment that Quantum is the first "direct sequel" Bond film - From Russia with Love explicitly references the events of Dr. No a couple of times.