Which variation of three knights game is best for Black

Pertaining to the line e4 e5 Nf3 Nc6 Nc3

You can learn the Steinitz Defense (which also works against the Scotch and may introduce you to some similar positions if you ever decide to pick up the Exchange Philidor) with 4… g6. This is a good option if you want more imbalanced positions where you accept passivity in the short term in exchange for equality and counterattacking chances in the long term. This would likely be my recommendation.

The Four Knights is your other option. This opening will often follow one of three paths: Scotch, Spanish, or Italian. In the Four Knights Italian you can choose either to play 4… Nxe4 5. Nxe4 d5, after which you’ve fully equalized with a very slightly better position, or you can choose to play 4… Bc5 or 4… Bb4 if you want to keep all of the pieces on the board and take a more positional approach in a closed position. The Four Knights Scotch is next, against which you can either play the Accepted with 4… exd4 or the Declined with 4… Bb4, both of which are fine options. The Accepted allows Black to equalize quite nicely but is rather drawish, and the Declined is better if you want more imbalance and tactical opportunities. Lastly, we have the Four Knights Spanish, which has a very large amount of theory should you decide to learn the Double Spanish or Rubinstein Countergambit, but 4… Bd6 works rather well as an off-beat option that retains good chances for Black. If White plays the 4. g3 Four Knights, just play 4… d5, 5… Nxd5, 6… Nxc3, 7… Bc5, 8… 0-0.

unfortunately, order matters. Trying to get a steinitz defense in the three knights variation with 4.g6 can get you in some trouble. The problem is subtle but significant. White has not "wasted" a turn developing his light squared bishop, which allows black to get a timely 0-0. (when white goes to b5, he must either lose a tempo after nxc6 or lose the bishop pair with bxc6, likelywise, if bc4, the move be6 is useful for black). 
without the bishop move, white can set up an immediate, nc3, be3, qd2 f3 0-0-0, and his attack is actually faster (this is actually discussed in great detail in the larsen variation chapter of Bauer's philidor file which almost refuted the variation , which it transposes to, but luckily, the larsen variation survives via the discovery of an early g6, nf6, ng4, ne5! before bg7) sadly, there is no equivalent variation in this move order.
chess is hard. actually, if you want to play something like a steinitz you should check out this post on an original sideline i "invented" https://www.chess.com/forum/view/chess-openings/rarely-seen-line-in-the-three-knights-opening