The three Modern Warfare games share a plot, but the plot tends to be pretty nonsensical. There are plot summaries online if you want to jump into MW3 without playing the first two.
As far as online goes, there are still many people playing the first two Modern Warfare games, although the majority of people are now playing Modern Warfare 3. (These are Xbox Live numbers, although you can assume PSN is similar).
The three games share a very similar engine and very similar gameplay. Skills from playing one online generally transfer to other games in the series.
I would say that you could almost consider Modern Warfare to be the "Madden" of shooter games - there's one every year, and the gameplay tweaks are generally of interest to the hardcore series player.
Don't forget also that in between Modern Warfare and Modern Warfare 2 was Call of Duty: World at War, and between Modern Warfare 2 and Modern Warfare 3 was Call of Duty: Black Ops. World at War and Black Ops don't share the same storyline as the MW series, but they still have similar gameplay and online components. They were developed by a different company (Infinity Ward does Modern Warfare, Treyarch the others). Some people have a preference for one company's work over the other's.
Which you choose is really up to you; price-wise the older versions are considerably cheaper, however, the bulk of the support and the gamers are likely to be on Modern Warfare 3. More players generally means more game lobbies, and a better likelihood of a good multiplayer experience, but your mileage may vary.