I play the Accelerated Dragon, which is kind of an universal response to 1.e4, and has the advantages of always being able to keep the positions into familiar territory for you: similar pawn structure, best squares for the pieces and many ideas are recurrent in different variations.
Against the Alapin (2.c3) Sicilian I suggest:
[fen ""]
1.e4 c5 2.c3 (2.d4 cxd4 3.c3 g6 4.cxd4 d5!) g6 3.d4 cxd4 4.cxd4 d5! {This position may also arise from the Smith-Morra gambit as shown above, so you are cutting off studying time! And it's completely sound.} 5.e5 (5.exd5 {This is the most challenging approach, but because of the position being not exactly the most played, advancing feels more natural.} Nf6 6. Nc3 Nxd5 {Some theorists suggest that this is hurrying too much to get the pawn back, but if you don't take it now you may end up struggling to get it back.} 7. Bc4 Nb6 8. Bb3 Bg7 {This is a sample line where black has driven the game into his territory and achieved an imbalanced position. A draw would be the most likely outcome though}) (5. Nc3 {This is a tricky sideline that's not much played, but worth knowing at least a few moves deep.} dxe4 6. Bc4 Bg7 7. Qb3 e6) {This is probably the move you'll face the most, so I'll expand more on it.} Nc6 6. Nc3 Bg7 7. Bb5 Nh6 {The knight goes to f5.} 8. Nge2 O-O 9. O-O Nf5 {Notice how the e pawn is not yet moved, you don't want to lock your bishop in.}
The main idea behind this move order is that white's interesting possibilities are highly restricted compared to others. If he goes for the most challenging approach, he would have to commit to an isolated d pawn, which is not usually the taste of 1.e4 players. If not, the'll have to play an improved version for black of the French Defense or try some "one-move threat" lines.
You'll most likely face opponents that advance you on e5. The main idea for black is trying to put loads of pressure on the weak d4 pawn. Moves like Qb6 are always to be looked at, specially since it could potentially be a double attack against also b2. This is why white's bishop is developed to b5, to avoid the double attack and to allow exchanging an attacker on said pawn.
As said in the comments of the game, you don't want to play e6 just yet. On some positions where black achieves the d5 advance, the Bc8 simply doesn't play. Now, he has an alternative route and on some variations it even goes to e6 itself. This is why this position is, for black, much better than the ordinary French Defense position that it may "transpose" to.
This said, the main ideas of attacking on the queenside still hold, and an interesting move to know is a well-timed f6. This little pawn break allows the Bg7 bishop to come into play with devastating effect, as the d4 pawn normally cannot stand the pressure. Of course, such a move has to be always played carefully, but if e5 would be possible in the near future, to take under control the whole center, it is definitely a good idea to consider it.
Conclusion:
If you like Dragon-like structures (or time is a big issue for you) and dynamic play, I encourage you to try this. The positions are essentially imbalanced, there is no queen trade in the near future and many tactical motifs typical from the Dragon are still playing an important role.
