So out of this opening black has generally will either take with 3 ... exd4 or transpose into the Philidor via 3 ... d6.

If they do take, is it best to go into the Goering Gambit via 4. c3!? dxc3 5. Bc4

Or to go into the Scotch Gambit via 4. Bc4?

Thoughts?

Except that no one has really had any success on a consistent basis with the Scotch except Kasparov and an incredible Russian I am not.

Seriously though, thoughts on the two gambits?

I would play 4.Nxd4 with a central pawn and knight and equal material. Did you see Yelena Dembo's win over Antoaneta Stefanova in the Scotch?

[Edit: OK, so you don't like the Scotch.] 

Well, after 1.e4 e5 2.Nf3 Nc6 3.d4 i think that 3...exd4 is the best move.

After 3...exd4 white has a few choices 4.Nxd4 which is the scotch game, 4.Bc4 scotch gambit and 4.c3 goring gambit.

Its interesting after 4.Bc4 as the opening is still not defined!  Blacks choices here seem to be 4...Nf6 which transposes to the two knights after 5.e5 d5 6.Bb5 Ne4 7.Nxd4 Bd7 (or Bc5) while 4...Bc4 and 4...Bc5 seem to play itno the scotch gambit.  4...d6 tranposes to a phillidor but note that the knight is on c6 not d7 if you play hanham variation*. Moves like 4...Qf6 and 4...Be7 seem to be more 'computer moves'.

I play the scotch gambit and most stronger players will go into the two knights. Weaker players tend to play lines involving ...d6 and although white does not get a crushing advantage he builds a nice centre. The lines against the scotch 4...Bc5/Bb4+ are most fun to play as white because of threats against f7. Like any open game playing the scotch gambit will teach you how important keeping the initiative is (just like in the danish gambit) because you cant afford to let black equalise in the opening. If you play 4.Bc4 as white you have a fair bit of theory to learn considering the transpositions that can occur, mainly the 2 knights.

If you want to base your white repitoire around the move order 4.Bc4 i recommend Lev Alburts book Chess openings for white explained as in this book a complete repitoire is based around that move order! There are many variations and to be honest, most black players who dont tranpose to the 2 knights are playing off the cuff so the biggest advantage you will get is a big centre usually +/= but you will get the occasional mate on f7.