I'd play Bc5 in hopes of transposing
Ponziani Opening

I'd play Bc5 in hopes of transposing
I respond with d4 (that's what computers say). It's actually stronger than it would be normally, because it attacks the bishop too. The d4 pawn is supported 3 times and attacked 3 times, so it's safe where it is. After the pretty much forced pawn trade on d4, white has a strong center with dual pawns, which is exactly the point of this opening, and black has to move the bishop (or continue exhanging and end up down material and position). Bb4+ seems like black's best option, and white has natural development from there with Nc3 against the check. Doesn't pay off for black.

Idea of c3 => playing d4 to establish good centre.
So Bc5 just plays in white's favor, allowing d4 with tempo won.
Black has to challenge white's plan, so d5 or Nf6 make the job, matter of taste.
I like to play d5 when I can in 1.e4 e5 so its my preference here.

I know that if white does play d4 I'm screwed, but I haven't encountered the Ponziani before in a game, so I never bothered learning the opening. I'd rather hope white is willing to transpose and if he doesn't, accept an inferior position.

Both 3. ... Nf6 and 3. ...d5 are the best moves.
Other moves such as 3. ...Bc5? 3. ...d6 3. ...f5?! 3. ... Be7 3. ... Nge7 usually give White a nice game.

When I first took up the Ponziani I had skipped the chapter on 3...d5, so that was effective for a month or two.
http://chessskill.blogspot.com/2014/02/losing-my-virginity-with-ponziani.html

Okay, now that I've looked into the opening more seriously, if I play Nf6 as black I'd play d6 to support the center, pin the knight, if the bishop is pushed back (h2-h3), I move it to d7, if the knight is pushed back (d4-d5) , to e7-g6. Then I'd play Be7 and castle kingside. Unforunately, this position is cramped and looks a bit passive. If I play d5 (which I like more, this'll likely be the line I play), I follow with Qd6 and Bd7 and castle Queenside, hoping to undermine white's pawn center.

annonymousHuman
There is a whole another thread on Ponziani Opening Traps with hundreds of examples.

@ AnonymousHuman :
after 4.d4 Nxe4 is one of the 2 moves, with exd4...
4...e5xd4 5.e5, then black has choice between Nd5 and Ne4, both ok for black.
About 4...Nxe4, white of course continues 5.e5 but then, once again, black has choice...
5...Cce7 is the main move, but Nb8 is supposed to be ok too, and those who love risky play can try 5...Bc5, an interesting gambit which is playable.
Let's go back to 5...Nb8, which u think is bad.
Wrong assessment, the main move 6.e5 leading to equality after Qe7! 7.Qd4 (what else, if Ne4 moves, there will be a nasty discover check...), where the white queen will be exposed in the centre. Often leads to an endgame after 7...d6 8.Qxe4 Qxe5, but 7...Nf6 is possible too...
About your move, 6.Be3, I've never seen it before your post, ok it's tricky but the moves you propose are just blunders. Who could play f6 really? d6 is more logical but Qa4 is usual tactics with Ne4 so advanced players will just develop the f8 bishop, i dont know which square is best, Bc5, Bd6, with an equal position in my opinion.
Thats the problem of this kind of trap, they lead to nothing if black is careful, but here playing f8 bishop is just the more logical way to continue development so there are few chances your opponent would play your f6 or d6.
I've started researching and I really like it. What would you guys play for black here? Think, there are some traps to be wary of...