Beginning graphics

I want to begin programming 2d graphics. The only graphics I've ever dabbled in were java applets (which barely count). I've practiced java for a year and a half, but I know enough c++ to get by.

My question: How does one go about getting into programming 2d graphics?

Do modern hardware cards even offer any acceleration for 2D operations any more, or is that gone with the old days of DirectDraw and the like?  I've noticed that DirectDraw hasn't even been included in the last few versions of DirectX.