I like to get my knight behind c4 and I don't mind exchanging that pawn, I just don't like it to get stuck behind my knight, therefore I often play the English opening.
Is there an established opening where both knights get behind c4 and f4 pawns? Even if that means exchanging those pawns with the d or e pawns? I mean, is there an opening where it goes like:
c4 Nc6, "or something", f4 and then later on Nf3 followed by Nc3
if so, then how well does that opening score?
There is the Four-Pawn-Attack against the King's Indian. I'm not sure how well it scores, but it has fallen out of fashion for some reason.
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1
1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.f4 O-O 6.Nf3
In general it is hard to reach this setup because black usually stakes a claim in the center as well.
Conceivably you would also get this setup in a Stonewall-Attack, if your opponent doesn't play c5, or maybe in a kind of Meran, where you suddenly throw in f4 instead of Nf3. But this is rather un-theoretical and presumably too slow to be good.
And of course after 1.c4 c5 you could go 2.f4. This has been played by a handful of players and there seems to be no immediate refutation.
The set-up with early c4, f4, Nc3, Nf3 can seldom be played successfully. Most often black's central pawns will stop this plan or chase away one of white's knights. For example 1. f4 d5 2. c4 d4 or 1. c4 Nf6 2. Nc3 e6 3. f4 d5 4. Nf3 d4.
The set-up can only be recommended if black refrains from putting pawns in the center, for instance by choosing a Dutch set-up themselves. In this case you can find games with strong players on the white side using this plan.
IM Gorbatov played a game where the white c- and f-pawns got exchanged for black's central pawns, like you speculated would happen:
[Event "Montecatini Terme (Italy)"]
[Date "1999"]
[Round "5"]
[White "Gorbatov Alexej (RUS)"]
[Black "Schwenk Andreas (GER)"]
[Result "1-0"]
[FEN " "]
1.c4 f5 2.f4 Nf6 3.Nc3 e6 4.Nf3 d5 5.e3 c6 6.b3 Bd6 7.Bb2 O-O
8.Be2 Nbd7 9.O-O Qe7 10.Nd4 Ne4 11.Bf3 Ndf6 12.g3 Nxc3 13.Bxc3
Kh8 14.Bg2 dxc4 15.bxc4 e5 16.fxe5 Bxe5 17.Qc2 Qc5 18.Qb3 a5
19.a4 Ne4 20.Bxe4 fxe4 21.Rxf8+ Qxf8 22.Rf1 Qg8 23.Nf5 Bxf5 24.Rxf5
Bxc3 25.Qxc3 Qe6 26.Rg5 Qf6 27.Rxa5 Rxa5 28.Qxa5 h6 29.Qh5 Qe6
30.c5 Kh7 31.Kf2 g6 32.Qh4 g5 33.Qh5 Kg7 34.Ke1 Qe7 35.Qg4 Kg6
36.h4 Qe5 37.Kf2 Qf6+ 38.Kg2 Qe7 39.Kh3 Kf6 40.Qh5 Kg7 41.hxg5
hxg5 42.Kg4 Qd7+ 43.Kxg5 Qd8+ 44.Kf4 Qxd2 45.Qe5+ Kg8 46.Qb8+
Kg7 47.Qxb7+ Kg8 48.Qc8+ Kg7 49.Qg4+ Kh8 50.Qh5+ Kg7 51.Qe5+
Kg8 52.Qe6+ Kg7 53.Qe7+ Kg8 54.Qxe4 Qd7 55.Qg6+ Kh8 56.Qh5+ Kg8
57.Qg4+ 1-0
A more high-profile game is the following, where white had some chances but over-pressed:
[Event "Sarajevo"]
[Site "Sarajevo"]
[Date "1960"]
[Round "6"]
[Result "0-1"]
[White "Bent Larsen"]
[Black "Petar Trifunovic"]
[FEN ""]
1.f4 f5 2.Nf3 Nf6 3.g3 b6 4.Bg2 Bb7 5.O-O e6 6.c4 Be7 7.Nc3
O-O 8.d4 Ne4 9.d5 Nxc3 10.bxc3 Na6 11.a4 Nc5 12.a5 Rb8 13.Nd4
Bf6 14.Ba3 Re8 15.dxe6 dxe6 16.Bxb7 Bxd4+ 17.cxd4 Nxb7 18.axb6
axb6 19.Qd3 Qd7 20.Rfd1 Nd6 21.d5 e5 22.Rf1 b5 23.c5 Nc4 24.c6
Qf7 25.fxe5 Rxe5 26.d6 Rd8 27.d7 Qe6 28.Qf3 Nd2 29.Qf4 Nxf1
30.Rxf1 Rxe2 31.Qxf5 Qe3+ 32.Kh1 Qxa3 33.Qg5 Qe7 0-1
Finally a historical game. White was one of the strongest players of the world at the time.
[Event "MacDonnell - Mackenzie 1862/63"]
[Site "London ENG"]
[Date "1862"]
[Round "9"]
[Result "1/2-1/2"]
[White "George Alcock MacDonnell"]
[Black "George Henry Mackenzie"]
[FEN ""]
1.c4 f5 2.Nc3 Nf6 3.f4 e6 4.Nf3 Be7 5.e3 O-O 6.Be2 d6 7.b3 b6
8.Bb2 Bb7 9.O-O Nc6 10.d4 Qd7 11.Qe1 Nd8 12.Rd1 Nf7 13.Qd2 Ne4
14.Qc2 Bf6 15.Bd3 d5 16.Ne5 Nxe5 17.dxe5 Nxc3 18.Qxc3 Be7
19.cxd5 Bxd5 20.e4 fxe4 21.Bxe4 c6 22.Qd3 Bc5+ 23.Kh1 g6 24.a3
b5 25.b4 Bb6 26.Bxd5 cxd5 27.Rc1 Rac8 28.Rcd1 Rc4 29.Qg3 Qf7
30.Bc1 Qf5 31.Rd3 Rfc8 32.Bd2 Rd4 33.Rff3 Rc2 34.Rxd4 Rxd2
35.Rdd3 Bf2 36.Qh3 Qe4 37.Qxe6+ 1/2-1/2
I play the Dutch with some variation:
[FEN "r1bqkb1r/pp2pppp/2n2n2/2pp4/5P2/1P3N2/PBPPP1PP/RN1QKB1R w KQkq - 2 5"]
This is the typical line I see. It gets very dangerous if you dont know how to defend against blacks counter attack. e4 if a very good move to prevent blacks d4 push.
Now the 4 pawn attack is extremely fun to play it is very hard to pull of if your opponent knows what they are doing. And it only ever works against the Kings Indian.
Back to your question, I don't think there is a common opening where you get BOTH knights behind your pawns but the English or the Dutch are both good opening to get either knight behind f4 or c4.
In addition to the openings that other users have already linked, the Four Pawns Attack in Alekhine's Defence can result in the two knights behind the c4 and f4 pawns.
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1
1.e4 Nf6 2.e5 Nd5 3.c4 Nb6 4.d4 d6 5.f4
The main line after the above position, however, results in black's d-pawn being traded for white's f-pawn, so the knight being behind the f4 pawn doesn't happen in most lines.
