I was massive fan of Pokemon back when I was a kid and played Pokemon Red extensively and completely understood the concept the locations and the 150 Pokemon. I also played Pokemon Gold but for the life of me the majority of Pokemon/Locations/Concepts/Storyline has not stuck with me because it has been such a long time.
I have recently got myself a Nintendo DSi XL and will start playing the games again but obviously since the last time I have played there have been a lot of new Pokemon games, story developments, locations, new Pokemon which is a little overwhelming.
1. No they don't continue between games. Well they, sort of, do between Black/White and Black2/White2. But they're basically separate stories.
2. As far as story elements go the only games to have chronology are Red/Blue - Gold/Silver and Black/White - Black2/White2.
The order of releases, though, is:
Ruby/Sapphire, FireRed/LeafGreen, Emerald.
Diamond/Pearl, Platinum. HeartGold/SoulSilver.
Black/White, Black/White 2
3. All the new games have new regions with differing climates and features. For instance; Pokemon Diamond, Pearl, and Platinum take place in the Sinnoh Region. It's a colder climate, with a lot of snow.
In Black/White and Black/White 2 they introduced seasons. So during Autumn the leaves are changing. In Winter it's snowing. Etc.
4. So far the only games to relate to Red are FireRed/LeafGreen (Remakes of the original Pokemon games for the GBA). Gold/Silver (And their remakes HeartGold/SoulSilver) allow you to travel back to the Kanto region. The only other way the games relate are that you can still find some of the original 151 in all the games, except the original Black and White.
5. The only game the show influenced was Pokemon Yellow. All the others influence the show.
The basic premise has stayed the same -
Collect Pokemon (Though, you don't have to capture them all anymore, you just have to battle against them for your Pokedex to register them.)
Follow the story lines surrounding the legendary Pokemon and the evil "teams" (Team Rocket, Team Magma/Aqua, Team Galactic, and Team Plasma).
Get 8 badges
Beat Elite 4
Any post game content
If I didn't explain things well enough, or if you have more questions, ask me. I hope you enjoy Platinum!