It is not 3.c3 and 3.d2, but 3.Nc3 and 3.Nd2.
The options here are different. Against 3.Nc3, you want to play a move that will put the question to e4. Either 3...Nf6, attacking it again, or 3...Bb4, indirectly attacking it by pinning the defender to the King.
You do not want to play 3...c5?? Because the center is not settled. You are, essence, trying to get White to push e5 to take all pressure off of d5 so that you can safely play ...c5, chipping away at d4.
With 3.Nd2, it is different. You cannot pin the Knight because White can answer 3...Bb4 with 4.c3! However, the Knight, on d2, is impeding White's development, and there is no pressure on d5. From c3, the Knight would attack d5. Here, there is no pressure on d5, and the 2 best squares for the Knight from d2, namely c4 and e4, are covered by the Black pawn on d5. So if d5 were to be isolated, it cannot be attacked easily, and it is an asset more than it is a weakness.
Therefore, Black should go after White's center immediately while his pieces are bottled up. Black should play 3...c5!, even though the center is not settled. Now 4.exd5 exd5 5.Ngf3, Black has a few options. With the White Knight eventually going to b3, both Knights will be 4 moves away from d5, and the d5 pawn is safe. Black's 3 main options are:
A) 5...Nf6 6.Bb5+ Bd7 7.Bxd7+ Nbxd7 8.O-O Be7 9.dxc5 Nxc5 10.Nb3 Nce4 with activity for the isolated pawn. Watch out for your soft spot - the f5-square - due to the removal of light-squared bishops. You will often play ...g6, even if the Bishop does not go to g7. Sometimes it does via f8.
B) The old main line - 5...Nc6 6.Bb5 Bd6 7.dxc5 Bxc5 8.Nb3 Bd6 9.O-O Nge7
C) My favorite - 5...Nc6 6.Bb5 Qe7+ 7.Be2 (7.Qe2 Qxe2+ 8.Bxe2 gets White nothing and 8.Kxe2 a6 and White must surrender the Bishop pair to Black) 7...Qc7 8.O-O Nf6 and now:
C1) 9...dxc5 Bxc5 10.Nb3 Be7 11.O-O O-O 12.Bg5 Be6 intending 13...Rae8 and in the long term, Black wants to battery his Rooks on the e-file.
C2) 9.Re1 cxd4 10.Nb3 (10.Bb5+ just helps Black develop) Bb4 11.Bd2 Bxd2 12.Qxd2 O-O with a near equal position. Maybe a teeny tiny microscopic edge for White.
Hope this helps. Any questions you have on specifically the French, feel free to ask me.
Hello, as a French player I come across either 3.c3 a lot and do see 3.d2 sometimes. I was trying to find a good way to play against each and I find many different variations that I could learn. I usually just improvise the Winawar variation but its really not good against the Tarrasch. What variation of the French should I learn? I can learn a different one for each one as well.