the only thing you're missing is historical reference, and anything important is explained within the game if you take the time to ask the people. A few references:
Mutants Mention "The Master". He was the original super mutant and the final boss of Fallout 1
The GECK. Eventually you'll go on a quest to find the GECK, which was the major McGuffin in Fallout 2
The Enclave and the Brotherhood of Steel both have a lot of history in the various games
Vault Tec and the various states of the Vaults are interesting even in Fallout 1/2
Harold the Mutant, who you can encounter in Fallout 3 has been in each of the Fallout games and has been a fan favorite through the years
Other then that, there's not really that much a lot of the names are the same, but they never played a huge role in the old games.
If you're interested in the older stuff, the best thing to do is either find a good Let's Play, or just look it up on http://fallout.wikia.com/wiki/Fallout_Wiki