If White opens e4 and Black replies c5, that's the Sicilian defense. Typical next moves are 2. Nf3 d6 3.d4 cxd4 4. Nxd4, etc.
Are there good variations for White where he DOESN'T play d4?
I've heard of 3. b4 (Wing Gambit), but I don't like gambits.
Does it make sense for White to play d3 instead, either before or after Bc4?
Two Sicilian variations that don't have an early d4 are:
Sicilian, Canal-Sokolsky (Rossolimo) Attack (B51)
[FEN ""]
1. e4 c5 2. Nf3 d6 3. Bb5+
Sicilian, Rossolimo Variation (B31)
[FEN ""]
1. e4 c5 2. Nf3 Nc6 3. Bb5 g6
I couldn't find much info on d3 before or after Bc4, but in some sample openings, d3 was played after Bc4 such as in the following:
[FEN ""]
1.e4 c5 2.Bc4?! e6 3.d3 d5!
It's quite common the Closed Sicilian, with, for example:
[FEN ""]
1.e4 c5 2.Nc3 Nc6 3.g3 g6 4.Bg2 Bg7 5.d3 d6
The Grand Prix Attack is of course another possible way to face any Black player after 1.e4 c5. Nowadays the main line is:
[FEN ""]
1.e4 c5 2.Nc3 Nc6 3.f4 g6 4.Nf3 Bg7
After 1.e4 c5 2.Nf3 (any) 3.g3 White can build a King's Indian Attack, which is less common and less sharp, but perfectly playable.
These variations are not as popular as the "Open" ones, but you may encounter them from time to time even during high-rated torunaments.
3.Bb5 is fairly common.
I personally like 3.Bc4; it just leads to a normal game with opening preparation thrown out. A plus or minus is that any type of position can result after the opening. 3.Bc4 sometimes shows up at the top-level, Vallejo beat Shirov with it, in this case the game resulted in a French structure.
[Fen " "]
[Event "SuperGM"]
[Site "Linares ESP"]
[Date "2002.03.09"]
[EventDate "2002.02.22"]
[Round "13"]
[Result "1-0"]
[White "Francisco Vallejo-Pons"]
[Black "Alexey Shirov"]
[ECO "B50"]
[WhiteElo "2629"]
[BlackElo "2715"]
[PlyCount "107"]
1. e4 c5 2. Nf3 d6 3. Bc4 Nf6 4. d3 e6 5. Bb3 Be7 6. O-O O-O
7. c3 Nc6 8. Re1 b5 9. Nbd2 d5 10. e5 Nd7 11. d4 Ba6 12. Nf1
b4 13. Ba4 Rc8 14. Bxc6 Rxc6 15. cxb4 Bxf1 16. Rxf1 cxb4
17. Be3 Qa5 18. g3 Rfc8 19. Ne1 Qb5 20. h4 a5 21. b3 a4
22. Nd3 Rc3 23. Nf4 Nf8 24. Qg4 Qd7 25. h5 axb3 26. axb3 Rxb3
27. h6 g6 28. Nh3 Rbc3 29. Bg5 b3 30. Rfb1 Rb8 31. Kg2 Rc7
32. Rb2 Bxg5 33. Qxg5 Qe7 34. Qe3 Qb4 35. Qf4 Qe7 36. Qf3 Rcb7
37. Rab1 Nd7 38. Rxb3 Rxb3 39. Rxb3 Qf8 40. Rxb8 Nxb8 41. Ng5
Nd7 42. Qf4 Qe7 43. Qc1 Qd8 44. Nf3 Kf8 45. Ng5 Kg8 46. Nf3
Kf8 47. Kg1 Qb8 48. Qa3+ Ke8 49. Ng5 Nf8 50. Qa4+ Ke7 51. Kg2
Qb7 52. Qa3+ Ke8 53. Qf3 Qe7 54. Qf6 1-0
The game at chessgames.com.
Vallejo did play 4.d3, so yeah it does make sense to play d3; it mainly allows the development of the dark-squared bishop.
When you get right down to it, white must move the d pawn sooner or later. It can either go to d4 (characteristic of the Open Sicilian) or it can go to d3 (characteristic of the Closed Sicilian).
With that said, on d3, the pawn blocks white's light squared bishop so white must make other arrangements in order to castle kingside. There are two main ways that this can be done - either white can move the bishop to c4 or b5 before moving the d pawn, or white can fianchetto the bishop.
So to bring this answer back to the specific question, white has a few concrete options that avoid d4:
3. Bb5 vs 2... Nc6 / 2... d6 respectively)1. e4 c5 2. Nc3 Nc6 3. f4 g6 4. Nf3 Bg7 and either 5. Bc4 or 5. Bb5)1. e4 c5 2. Nc3 Nc6 3. g3 g6 4. Bg2 Bg7 5. d3 d6)Surprisingly, almost every other Sicilian variation for white either involves white playing an early d4, or is very similar to one of the above. For example, the King's Indian Attack is really a variation of the Closed Sicilian (alternatively, the Closed Sicilian can be considered a variation of the KIA...).
Finally, the Wing Gambit merely defers d4, as does the Alapin. While white can always choose to omit d4 (2. Qh5?! would be an extreme example), these are generally considered dubious sidelines played at white's own peril.