The short answer is to start with this game.
The long answer is that the first Legend of Heroes game was the sixth mainline Dragon Slayer game (not counting sequels and spin offs of the previous mainline titles). LoH then proceeded to follow tradition and got a sequel with LoH 2 and a stand alone trilogy with LoHs 3-5 (Prophecy of the Moonlight Witch, A Tear of Vermilion and Song of the Ocean respectfully) forming what is otherwise known as the Gagharv Trilogy. Then in 2004 LoH 6 Trails in the Sky came out for the PC in Japan and in three parts. The first part (LoH6:TitS FC) is the game this thread is posting about (its only available on the PSP in the west, and this thread is about an updated version of the original PC release getting released on Steam), and the second part (LoH6:TitS SC) wraps up the story started in the first part and is what everyone who is familiar with this series is waiting for (it didn't come out for the PSP in the west due to the fact that the PSP was very much a dead system outside of Japan, hence the big effort to get both parts out on Steam now). The third and final part (LoH6:TitS The3rd) mostly reopens the setting to follow a bunch of side characters and leaves enough plot threads and setting details hanging for following up on with later games. It's from there that Trails in the Sky spun off into its own series with Trails in the Zero and Trails in the Blue (still referred to Legend of Heroes but no longer numbered as such) forming a duology, and currently Trails in the Flash which is getting its sequel, Trails in the Flash 2, in about two months in Japan. There's also Nayuta in the Sky which is largely its own separate thing, and Ys vs. Trails in the Sky: Alternative Saga which is a crossover/mashup of Ys and Trails in the characters in a fighting game (built using Ys Seven's engine) just to make things even more confusing (and because Ys and Trails in the Whatever have become Falcom's two biggest series thus far).