Since the expansion packs contain both question and answer cards, you certainly can play with just that. You'll just start recycling cards sooner due to the limited number. You can download the rules from the CAH website (http://cardsagainsthumanity.com/).
You can buy the expansion decks only. Alternatively, you can download the original for free and get the instructions and buy the expansion decks and use those. Then you can choose to get the original deck printed out yourself - it is much cheaper that way, but the cards aren't quite as nice - you can still play that way, though
You don't necessarily have to by the original, but it doesn't make much sense because the amount of cards you get in the expansion is very small compared to the core game. Also, like others have said, its nice to have the instructions. If you can't buy them all, I know in the past they have provided the ability to print the cards for free. I think its worth buying them to support the company, but also the quality of the cards would be much worse if you printed them off.