I was introduced to the game by friends who owned Base but not Intrigue, and so the first box I bought on my own was Intrigue. That meant I had a number of games with family with Intrigue but no Base cards. It wasn't bad, per se, but not having the Base set available really makes you realize how important those cards are, conceptually, to the game of Dominion. They really represent the essence of the game.
So, I would recommend the Base set as one of your three. Intrigue is a great compliment to the base set, but I find it doesn't change the game that much. I love playing with Intrigue, but playing with just Base and Intrigue wasn't substantially better than playing with just Base. Consequently, I would recommend doing another set instead of Intrigue.
Everyone loves Seaside, so that's a solid recommendation. I have yet to find someone who didn't enjoy that set.
So, you've got Base and Seaside. Prosperity has some of the most powerful cards in the game and makes killer engines possible, so that's a strong contender for your third set. Cornucopia is fun but complex, and with only half as many cards it's not a great choice if you're just buying three sets. Hinterlands is one of my favorite sets because it really adds some strategic depth, and it does pair nicely with Seaside as the "this turn/next turn" thematic combo, so that would be a second reasonable choice. I like Alchemy a lot, but it's not a great choice if you are playing with newer players.
Thus, my recommendations would be Base/Seaside/Prosperity, followed closely by Base/Seaside/Hinterlands.
