How do I combat this opening by white (1.e4 - e5, 2.bc4)

I have played a bunch of games tonight, and have run in to this opening far more often than usual. King's pawn forward, I do a move and then white square bishop targets f2, and you just know the queen is coming out to play in the immediate future.

Here is a game I played (as black naturally), and this one turned out OK:

However I feel like he made some odd moves and I was lucky. Can anyone tell me what the correct way to counter this is?

The way you played it is OK. 

This is the Bishops opening. I like to respond with 2...Nc6 in order to avoid an immediate 3.d4 exd4 4.Nf3 which gives White a slight initiative. After that depends on Whites third move 3.c3, 3.d3 , 3.Nf3 I continue develop towards the center. Furthermore 2...Nc6 allows to meet an early Qh5 with g6 and in some lines you may consider Na5 to trade it off with the light square bishop.

Thanks for that info @Bishop_g5, I like the idea of the Knight move first, and avoiding the middle pawn exchange. I think that's where the trickery comes from for me, as I often pull my Knight away which is preventing the Qg4/h5 move. Nice!

Yeah this one worked out. But I do think I was lucky, the last few moves he played really quickly - all under 10 seconds and I don't think he calculated properly. But I've had games where I've been the one to do that, and dropped a Knight or something. I do like Bishop_g5's move order though.

Hey guys! How about playing 2 ... Qh4 ??  The game I tried this in is as follows: 

Pretty good for black!

The OP could have a look at Bu's games against Carlsen and Svidler. Think they are annotated a bit on chessbase site.

As someone else mentioned, 3...d5 is questionable. Of the 69 master games that continued 4 exd5, White won 46, Black won 10, and 13 were draws.

Back in the '90's, when the Bishop's Opening had been out of favor for 60 years, the high school team I coached researched it, practiced it, and rode it to three straight county titles and three straight State Scholastic Championship team trophies.

3...d5 didn't worry us.  3...Bc5 gave us the hardest time: our goal was to play 4 Nc3 and 5 f4, transposing to a favorable-for-White position in the King's Gambit Declined, before playing Nf3.