What's the best continuation for white here? Here's what I do when someone tries to copy my moves:

[Date "????.??.??"]
[Result "*"]
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 Qe7 5.Qxe4 d6 
*


But then my opponent plays d6, and I don't know what should I do. What's the best continuation for white after d6?
You're not down material, so just develop.

6.d4 allows both your bishops to get in the game while keeping strength in the center. After that, you can develop your minor pieces and castle very efficiently. You're going to have a swell game.

I wouldn't worry too much about 7... f6. That move weakens his Kingside. And since you're up a pawn, the best he can do is retake on e5 and give himself an isolated pawn. 8. Bf4 develops a piece and puts an eye on the under-defended c7 square.

Furthermore, after Bf4, you can plan for Nc3 followed by O-O-O (castling on the queenside) where your rook will thunder on the d file.