Hi EasternRegent,
I have played against this opening a number of times, and the move that I prefer to play is this:
However, it is very difficult to give you a complete repertoire because there are so many different moves that White can play after this. What move(s) are you struggling against the most?
I too have played Nf6, but I still seem to have trouble taking the pressure off f7 and creating pressure of my own. I tend to lose my center pawns in the process. I will try to find the last game I played against it and post it.
I too have been seeing the Bishop's opening more and more.
Looks to me like White has to defend e4. After they do so, you could play either d5 or d6 and then Bf6. But then it has been a few years since I played the Bishop and then it was as white. I don’t see the Bishop as Black because I play the French.
After 3. d3 it's possible to play in the centre with c6+d5.
That helps.
I have been seeing the Bishop's Opening more and more and would like to know how to best play against it. I seem to get off to a bad start and have to work back to a good state.
just follow opening principles:
Control the center
Develop minor pieces toward the center
Castle
Connect your rooks
I tend to see 3. Nc3 more than 3. d3, which transposes the game into an Italian or Vienna more often than not. There is a lot of theory on those openings that you could study.
Good luck! 
Here is a game with the line pfren mentions after Nc3. I think Caruana had at least some advantage but that probably wasn't because of the opening which is quite equal but because he played better later on.
Possibly helpful:
Beating the open games, Mihail Marin
https://web.archive.org/web/20140626195205/http://www.chesscafe.com/text/hansen102.pdf
Open-Games-Black-Igor-Lysyj
https://web.archive.org/web/20140627104938/http://www.chesscafe.com/text/hansen159.pdf
Play the Open Games as Black by John Emms
https://web.archive.org/web/20140627022601/http://www.chesscafe.com/text/hansen11.txt
Bologan's Black Weapons in the Open Games
http://www.jeremysilman.com/shop/pc/Bologans-Black-Weapons-in-the-Open-Games-76p3873.htm
Starting Out: Open Games by GM Glenn Flear (2010)
https://web.archive.org/web/20140626232452/http://www.chesscafe.com/text/hansen134.pdf
Play 1 e4 e5! by Nigel Davies (2006)
https://web.archive.org/web/20140626201436/http://www.chesscafe.com/text/hansen80.pdf
https://web.archive.org/web/20140627083715/http://www.chesscafe.com/text/hansen98.pdf
The Kaufman Repertoire for Black & White by Larry Kaufman (2012)
https://web.archive.org/web/20140626221508/http://www.chesscafe.com/text/hansen162.pdf
Opening Repertoire: The Open Games with Black by Martin Lokander (2015)
Playing 1.e4 e5 - A Classical Repertoire by Nikolaos Ntirlis (2016)
3...c6 is definitely the critical line.
I don't know what you usually play but against me almost everyone plays ...Nc6/...Bc5 instead, which basically leads to a favourable King's Gambit for White after f4.
If I'm not mistaken Bareev played 3...c6!, he lost that game, I bought the chess informant back then, it was light blue (57 I think), I can't find it right now, but anyway the opponent was Kasparov. Played in the past by Greco, Lucena, Philidor and Bent Larsen ( but don't take this for granted ).
I have been seeing the Bishop's Opening more and more and would like to know how to best play against it. I seem to get off to a bad start and have to work back to a good state.