Pokemon: Gotta catch 'em all, but how to get started?

I don't think there's anything lost by not playing previous generations; just buy the newest games. All the newest DS games (Diamond, Pearl, Platinum, HeartGold, and SoulSilver) can connect to each other.

The battle mechanics were hugely overhauled after the "Advanced Generation" (Ruby, Sapphire, Emerald, FireRed, LeafGreen), so I would not recommend starting with those games. If you do, it will take a little time to adjust when you start playing the DS games.

Pokemon Red, Blue, Yellow, Green, Gold, Silver, and Crystal cannot connect to the newer games, so if you think you will become attached to the Pokemon you catch and train in them, it's not a good idea to start with them. Pokemon from the GBA games can be "migrated" to the newer DS games, but you need to have a DS with both a DS and GBA slot (the original DS and the DS Lite).

In short, I suggest you start with either HeartGold or SoulSilver, as those are the newest games.

If you don't want to play the first gen, but still want to not miss anything you could start out by playing the third gen remakes of the first Gen: Fire Red and Leaf Green. Then go to the much newer HeartGold and SoulSilver, then back to Ruby/Sapphire/Emerald, then Diamond/Pearl/Platnium, Black/White, Black 2/White 2, X/Y, etc. This will progress with the original game order only substituting the first two's remakes in for them. The disadvantage with this is, that with HG&SS in there, you'll be going from GBA graphics, to DS, to GBA, to DS, along with the battle systems changing between the third and fourth game, so you might want to play RSE before that.

Although, considering playing all the game would take a long time, and they're pretty separate, I'd probably recommend starting with HeartGold or SoulSilver then go to Diamond, Pearl, and Platinum, or maybe just start with Platinum.(I'd recommend it because it has more features, better graphics, and stuff than D&P.)

Because you haven't had a lot of experience in the series, it is not as if you'll have problems adjusting to mechanical changes implemented on later generations, like the physical-special split. Nor would you miss any of the special features that were in earlier versions. All of the games are isolated in their stories and walk you through everything, so you don't have to worry about getting lost.

Assuming you have the right platform, the best idea in general is to play whichever generation is the one you expect to interact with other people in. While the game is perfectly serviceable alone, you get a lot more out of your playtime if you have someone else to game with, and that'll only work if their generation matches yours.

This is generally going to be the latest generation, since that's what most people will be playing at the time. It's also the generation which is most likely to still have official events going on, meaning great chances to get the event-only Pokemon. Starting 4th generation, it's also possible to interact across the world, which means you don't even need friends in your immediate vicinity. Also, later generations have more Pokemon which generally means a lot more options for team experimentation.

You will not miss any "lore" by avoiding the earliest games. The majority of the games are each in a self-contained region, and while your actions in the game usually involves saving that area from a criminal organization known as a "Team" (e.g. Team Rocket, Team Magma, Team Galaxy, etc.), none of the other regions refer to any of the others, other than possibly containing "foreign" pokemon.

Ironically, the one game pair in the series, Gold/Silver, that does refer to previous areas, is the most recent version of the game (Heart Gold / Soul Silver), but even this is only in passing, and only apparent if you're familiar with how things were in the region previously. (And not highly relevent. Example -- the Safari Zone is now closed because the warden is on vacation.)

The bottom line? Pokemon is not popular for its plot; don't let that determine which version to buy. If you're just starting with the series, the most recent games are as good of a place to start as any.

To further contribute to Raven's comment, if you wish to appreciate the older graphics and such, you would probably want to start at Diamond/Pearl/Platinum, as they're the last game versions where the graphics where mostly 2D. Otherwise, you can start anywhere, even the latest ones which are currently generation 6 games (X, Y, Alpha Saphire, Omega Ruby), because I've heard all the pokemon (except mythical ones) can be obtained from those few versions and you will be caught up in terms of pokedex completion. However, if you wish to follow the lore, then you're in for a rough time! There's too many versions to play to catch up, unless you speed run them with one or a few pokemon