Black's most critical test of the gambit is to accept it, and this approach also offers Black the greatest winning chances, if he/she knows what he/she is doing. However, it also gives White what he/she wants, and gives White the majority of the piece play and attacking chances, and the resulting positions often become complicated very quickly.
White's most usual plan is then to put pressure on f7 with Bc4 followed by Qb3 and/or Ng5, and Ng5 can often be followed up with f2-f4, adding more control of the centre and preparing to push forward with e4-e5 and/or f4-f5. Meanwhile, Black tends to have difficulty developing the kingside pieces, and a common theme of this gambit is that the black king often ends up stuck in the centre into the early middlegame.
White's safest option is to recapture on c3 immediately, settling for a one-pawn gambit. This line is probably fully sound, but Black has the option of entering good lines following 5...Bb4 and 5...Bc5 that cannot be reached by force if White offers a second pawn with 5.Bc4. Black's first major decision is what to do with the dark-squared bishop.
5...Bb4 has traditionally been regarded as the most critical response. Black pins the knight on c3 to the king and intends to swap it off, and play ...d6, ...Nf6 and ...0-0 with a solid position. The usual continuation is 6.Bc4 d6 (Black can also flick in 6...Bxc3+ and then 7...d6, but in other cases, ...Nf6 tends to get hit by e4-e5).
7.0-0 (unpinning the knight on c3) appears to provide insufficient compensation after 7...Bxc3!, exchanging off the c3-knight before it can hop into d5. 7.Qb3 is an improvement, but even here, it is not clear if White can get full compensation for the pawn after 7...Bxc3+!. However, I believe that White can get enough compensation by playing Bc4 followed by Ng5 (instead of 0-0 or Qb3). This prepares the f2-f4-f5 pawn push as well as attacking f7 immediately.
An interesting alternative option for White at move 6 is 6.Bg5, but a closer look suggests that it's not convincing.
5...Bc5 is a tough and generally underrated defence. Here the traditional attacks against f7 don't work very well, for reasons that are not immediately obvious (essentially, Black's control over the d4, e3 and f2-squares works in Black's favour). However, White can take advantage of the bishop vacating the kingside by playing 6.Bc4 d6 (again, 6...Nf6 is hit by 7.e5) and then 7.Bg5, taking advantage of the dark squares around the black king. 7.Bg5 also keeps open the option of castling long, although this plan is often unconvincing and there is a strong argument that 7.0-0 followed by 8.Bg5 may be more precise as it narrows down Black's good options.
5...d6 prepares solid development with ...Nf6, ...Be7 and ...0-0. This line also commonly arises after 5.Bc4 (5...d6 6.Nxc3). White's only way to get enough compensation against this is to attack f7 immediately with 6.Bc4 followed by 7.Qb3 or 7.Ng5. The main line runs 6.Bc4 Nf6 (consistent and best) 7.Qb3 Qd7 8.Ng5 Ne5 9.Bb5 c6 10.f4, with general board-wide tactical chaos, and approximately equal chances for both sides. White has to be careful of the ...Nc6-a5 "fork" in this variation, which can force the exchange of the bishop on c4 for the knight on c6.
5...Nf6 6.Bc4 should be followed up with 6...d6, transposing to 5...d6, since other moves are well met by 7.e5.
Other moves should not trouble White. 5...h6 is a common reaction at club level, but White can get more than enough compensation for the pawn by continuing with Bc4, Qb3, 0-0 and then preparing e4-e5; even getting in the ...Nc6-a5 "fork" tends to fall short for Black, due to being a long way behind in development. 5...g6 is tricky because if 6.Bc4 d6 7.Bg5, exploiting the weakened kingside dark squares, Black can get away with 7...Be7!?, but Mark Nieuweboer's suggestion of playing 7.0-0 first, waiting for Black to commit the knight on g8 to f6 or e7, and then playing 8.Bg5, looks very convincing. If 5...Bd6, suggested by Siegfried Kalkofen and pointed out to me by Stefan Bucker, White should continue with 6.Bc4 and either 7.Bg5 or 7.0-0, with the idea of pushing e4-e5, and if Black puts a piece on e5, White should exchange it off and play f2-f4, gaining time and space.
Practical results In the Chesslive.de database as of February 2017, White is scoring 56% with 5.Nxc3 (as opposed to 62% with 5.Bc4). Black is scoring 45% with 5...Bb4, 46% with 5...Bc5 and 41% with 5...d6. Those stats don't tell the whole story, though, for after 5...d6 6.Bc4 Nf6 Black is scoring 52%. I tend to think that all three moves are of roughly equal value and that the choice is a matter of taste.
The most common way for White to continue, however, is to sacrifice a second pawn, this time on b2. The idea is that after 5...cxb2 6.Bxb2 the bishop gets to the long diagonal, giving two raking bishops pointing at the black king, and supporting an e4-e5 push and hindering Black's kingside development further.
If Black wishes to decline the second pawn, 5...d6 and 6...Nf6 is the best way, leaving White with nothing better than 6.Nxc3 transposing to the 5.Nxc3 d6 6.Bc4 Nf6 lines discussed above. Black cannot get into the preferred lines with ...Bb4 or ...Bc5, because 5...Bb4 can be met by 6.0-0 or 6.bxc3 (both of which improve over the 5.Nxc3 Bb4 lines for White) and 5...Bc5 can be met by 6.Bxf7+, followed by Qd5+ and Qxc5.
Most theoretically critical, however, is 5...cxb2 6.Bxb2. White has an increased initiative and development lead, but is it worth two pawns? I'm not sure, but in practice White scores better than average (61%).
If Black wishes to use the bishop on f8 actively, 6...Bb4+ is the only good way (other bishop moves are met by 7.Bxg7). The late USCF master Mark Morss favoured this line in correspondence play and got good results with it. The main line runs 7.Nc3 Nf6 8.Qc2 d6 9.0-0-0 0-0 10.e5, with a very dangerous initiative for White, although objectively the position is somewhat unclear. I am not, however, as convinced by White's chances if Black chooses to exchange off a couple of pairs of minor pieces with 9...Bxc3 followed by 10...Be6. However, it seems that castling kingside is a reasonable alternative for White and also gives a dangerous initiative.
Most solid for Black is 6...d6 which has been recommended in various sources (e.g. Batsford Chess Openings 2, John Emms's Play the Open Games as Black, and John Watson and Eric Schiller in Survive and Beat Annoying Chess Openings). It consigns Black to a passive position, but Black remains two pawns ahead and makes it difficult for White to breach the black defences. White's three main options are 7.0-0, 7.Qb3 and 7.Nc3. The 7.0-0 line has a well-established main line which gives White continued pressure into the middlegame but it is not clear if White gets enough for two pawns. I have a slight preference for 7.Qb3 because it attacks f7 immediately and gives Black more scope to go wrong. 7.Nc3 intends 8.Nd5 and also looks fully playable.
While 5...cxb2 is probably objectively the best move, in the Chesslive.de database Black is scoring just 44% with 5...cxb2 6.Bxb2 d6, 39% with 5...cxb2 6.Bxb2 Bb4+ and 52% with 5...d6 6.Nxc3 Nf6, so, especially in over-the-board games, the players who prefer to decline the second pawn with 5...d6 have a strong case for their position.
In summary, the Göring Gambit appears to be reasonably sound. It does not promise a theoretical advantage, but in practice White, overall, has a greater than normal plus score (60%) in the accepted lines of the gambit. It is primarily the 4...d5 declining variation (which scores a respectable 48% for Black) that puts many players off from using the gambit.