Runaway: Dream of the Turtle takes place just after the first game.
A Road Adventure was told in flashback format by our main character, Brian Basco. His first adventure starts off with him traveling across country after getting his Physics degree. All is well and good until he hits a dancer named Gina Timmins and the two are forced to dodge the mafia, uncover ancient Hopi secrets and, of course, fall in love. That's really all you need to know about the previous game's plot in order to play
Dream of the Turtle, but I would definitely recommend picking up and playing through that game before starting on this second adventure.
In the span between the games, it seems like Brian's demeanor has changed quite a bit. He is no longer the prim-and-proper academic. Instead, he has come out of his shell a bit and is exploring his inner surfer-dude. While I can't say I completely like the change in style, I can only assume it has more to do with the Hawaii-based locale of the game than an actual change in Brian's character since he is still smart. He is just a bit more outgoing than before.
So, Brian and Gina are having some fun in the sun in Hawaii, when the two characters find themselves in a free-falling plane. Brian forces the only parachute on Gina and shoves her out the door while he holds on tight for the impending crash. And that's just the opening sequence.
Stranded in the middle of a lush jungle, Brian must find Gina and get them back to safety. But on the way, the duo will find themselves in The Temple of Tiki, communicating to aliens, and even doing some minor globe-hopping.
Some of the better parts of this game are when side characters from the previous title make guest appearances. It is always fun to see how the game carries over aspects of its prequels, but still doesn't rely wholly on the player's knowledge of who that character is or was. This makes Dream of the Turtle accessible for players who have been through the original story and those who haven't. On a side note, Runaway comes with a short video to quickly recap the people and events of the first game, but you don't get nearly the impact and feeling of who those characters are unless you play through A Road Adventure.