I fully agree with Ptero. If the main goal is improving, then long time controls - maybe even corr chess - is absolutely necessary. I would also say, that it is important to play different types of openings. It is even strongly to be considered, to incorporate one gambit - say the Marshall Gambit 1.d4 d5 2.c4 e6 3.Nc3 c6 4.e4. The idea of course is to get an allround development and work hardest on the weakest sides.
But if you want to stick to blitz and rapid, just play what you like best.

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I don't intend to remain on a classical repertoire, I just thought that learning to play those openings decently might make me a better chess player. Certainly I will have to face positions more similar to the classical openings than any Dragon and I need to be able to play those as well as their evaluation would suggest. Right now, I'm a lot like thibdb13 and do not handle those positions as well. However, I believe that hypermodernism and classical positions are both valid and so I should be able to play both well. Moreover, I should not be afraid of moving into more classical positions just because they are not what I'm used to. It's just something I need to work on.

Also, on ICC, I play both blitz (8 min 10 sec increment) and standard (20 min 20 sec increment or more). However, I prefer standard. Blitz is fun too, but it's like going to McDonald's instead of a good restaurant.

BTW: thibdb13, what openings do you play vs d4 and e4?

Classical openings can be interesting to follow: they allways contain ideas (thematic moves) that might be usefull in other openings. But I think it is very difficult for someone playing hypermodern to adapt to the classical openings. Personnally I do not feel comfortable at all when answering with e5 on e4 and usually lose these games even if I was not worse in the opening: I am just not good in these situations Maybe the philosophy is too different

I started with the Hyper-Accelerated Dragon and Nimzo/Bogo (but odd and not that great Nc6 variation) when I started playing chess more seriously, almost three years ago. Thus, I know next to nothing about the Slav, QGD, or Ruy Lopez. Now, being a little over 2000 on ICC, I'm wondering if I should still learn and play these more classical openings to get a better chess education. 

I don't know if this matters, but I am neither tactical or positional but rather a balanced mix. I just try to play the best moves and focus on activity, dynamics and initiative. Moreover, I seem to be better in hypermodern positions where my center is not committed and many pawn breaks are possible. I am also quite good in positions where wing play is essential, and the center is more or less under control, such as in the English attack from the Najdorf. Thus, I may have some weaknesses in fully utilizing or fighting against a classical center with the associated piece configurations. (for example a Scotch game) Although, I have improved in this area since learning the French. Would classical openings help? What should I do?