I'm only the beginner but I believe the 3. ...a6 is strongest, it's good to clear what this bishop wants before doing something else. And then... I think Chigorin system and modern Steinitz is the best to study first to understand strategies of this opening, and after this you can try another strong a6 systems to find your own.
Which opening you play versus the Ruy Lopez?

First I played for a while scandinavian defence, switched to icelandic gambit, then tried to get basic 1.e4 e5 2.Nf3 Nc6 and go from there.
Now days I'm trying to learn Alekhine defence.
You're in the wrong thread

You wouldn't be able to play a Modern Steinitz or Breyer against me. I'll go main lines against stuff like the Berlin, Classical, 3...f5, Steinitz, Bird's, etc.
However, those that play 3...a6, I'm taking the Knight. No Breyer, no Zaitsev, no Chigorin, no Modern Steintiz, no Moller, no Archangel, etc.
The exchange is a good choice imo and certainly limits what you have to know . I have a good friend that plays only the exchange against 3...a6 for decades now and I am pretty sure you know him too Thriller ... I sometimes play the exchange as well but more often play 4 Ba4 and have a pet line I try to get into around move 5/6 as white .
I play both sides of the Ruy so besides defending with the Modern Steinitz, I do play the Exchange variation myself, but with 5. Nc3.

I am somewhat interested in 3...Nge7 and 4...g6. I am not an e5 player (at the moment) but if I build a repertoire I will play that Fianchetto system.
I am somewhat interested in 3...Nge7 and 4...g6. I am not an e5 player (at the moment) but if I build a repertoire I will play that Fianchetto system.

3...g6 (the Smyslov) is a good defense.
The Cozio (3...Nge7) has an issue against 4.Nc3, when 4...g6 is quite dubious, and Black IMO has to transpose to a Steinitz after 4...d6.
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 Ruy Lopez: A Guide for Black by Sverre Johnsen and Leif Johannessen (2007)
https://web.archive.org/web/20140627023224/http://www.chesscafe.com/text/hansen94.pdf
A Spanish Repertoire for Black by Mihail Marin (2007)
https://web.archive.org/web/20140626195205/http://www.chesscafe.com/text/hansen102.pdf
The Berlin Wall by John Cox (2008)
https://web.archive.org/web/20140627050257/http://www.chesscafe.com/text/hansen115.pdf
Attacking the Spanish by Sabino Brunello (2009)
https://web.archive.org/web/20140627105306/http://www.chesscafe.com/text/hansen124.pdf
The Ruy Lopez Revisited by Ivan Sokolov (2009)
http://dev.jeremysilman.com/shop/pc/Ruy-Lopez-Revisited-The-p3765.htm
Fighting the Ruy Lopez by Milos Pavlovic (2009)
https://web.archive.org/web/20140627105306/http://www.chesscafe.com/text/hansen124.pdf
Slay the Spanish by Timothy Taylor (2011)
https://web.archive.org/web/20140627040230/http://www.chesscafe.com/text/hansen146.pdf
The Kaufman Repertoire for Black & White by Larry Kaufman (2012)
https://web.archive.org/web/20140626221508/http://www.chesscafe.com/text/hansen162.pdf
The Berlin Defence by Igor Lysyj & Roman Ovetchkin (2012)
https://web.archive.org/web/20140627104938/http://www.chesscafe.com/text/hansen159.pdf
Grandmaster Repertoire 13 – The Open Spanish by V. Mikhalevski (2013)
http://reviews.dailychess.org/grandmaster-repertoire-13-the-open-spanish-qualitychess-by-v-mikhalevski/
Anti-Spanish. The Cozio Defence by Alexey Dreev (2014)
Bologan's Ruy Lopez for Black by Victor Bologan ("Coming soon")
https://www.newinchess.com/Bologan_s_Ruy_Lopez_for_Black-p-9016.html

3...g6 (the Smyslov) is a good defense.
The Cozio (3...Nge7) has an issue against 4.Nc3, when 4...g6 is quite dubious, and Black IMO has to transpose to a Steinitz after 4...d6.
I learned that 3...Nge7 served to avoid 3...g6 4 d4 exd4 5 Bg5 after Black ''stands awkward'' as that person said. Is that alright for Black, though?

3...g6 (the Smyslov) is a good defense.
The Cozio (3...Nge7) has an issue against 4.Nc3, when 4...g6 is quite dubious, and Black IMO has to transpose to a Steinitz after 4...d6.
I learned that 3...Nge7 served to avoid 3...g6 4 d4 exd4 5 Bg5 after Black ''stands awkward'' as that person said. Is that alright for Black, though?
Certainly better than 3...Nge7 4.Nc3 g6 5.d4 exd4 6.Nd5! when Black can't equalize.
You can compare with the analysis at Dreev's book on the Cozio. 16.0-0-0 is a novelty, although an obvious one. Black is about to lose a pawn for no compensation...

You wouldn't be able to play a Modern Steinitz or Breyer against me. I'll go main lines against stuff like the Berlin, Classical, 3...f5, Steinitz, Bird's, etc.
However, those that play 3...a6, I'm taking the Knight. No Breyer, no Zaitsev, no Chigorin, no Modern Steintiz, no Moller, no Archangel, etc.
The exchange is a good choice imo and certainly limits what you have to know . I have a good friend that plays only the exchange against 3...a6 for decades now and I am pretty sure you know him too Thriller ... I sometimes play the exchange as well but more often play 4 Ba4 and have a pet line I try to get into around move 5/6 as white .
Pet line on move 5 or 6? Sounds like the Qe2 lines. The Worrall Attack if I remember correctly. I own the book on it, and played it once, and played the main lines as Black a time or two, but mostly the Worrall is a non-factor for me as the Exchange as White avoids that and when I was playing 1...e5 as Black, while I have played many lines, I mostly played the Berlin, and specifically the Endgame Variation (5...Nd6) and not the Rio DiJenaro (5...Be7).
Can't say I'm the guru of 1...e5 as I've been playing 1...c5 exclusively for over a year now, specifically the Najdorf and Taimanov. No pattern or frequency. Whichever I'm in the mood for at that time.

The Berlin Defense is a great way to lose as Black if you're rated U2000. This is borne out by the results, and it makes sense because Black is striving for a very difficult position that even Super GMs misevaluate.
For most players, I really like the Jaenisch/Schliemann Defense with 3.f5!? It's sharp, relatively unknown, and fully playable all the way up to +2800 ratings!
Another good opening is the Marshall, if your opponent will let you get that. Below ~2300, it's not very drawish.
A third good choice, and one that Reb and I agree on, is the Breyer. It's an excellent thematic opening that rewards the player who understands the ideas behind the opening best.

Here's one example of the Schliemann Defense where White was one of the great experts on the Spanish, Mickey Adams:
And here's one between two of the best players in the world:

3...g6 (the Smyslov) is a good defense.
The Cozio (3...Nge7) has an issue against 4.Nc3, when 4...g6 is quite dubious, and Black IMO has to transpose to a Steinitz after 4...d6.
I learned that 3...Nge7 served to avoid 3...g6 4 d4 exd4 5 Bg5 after Black ''stands awkward'' as that person said. Is that alright for Black, though?
I think that might be an outdated opinion. Black seems fine after the anti-thematic 5...Be7 as played by Azmaiparashvili. Huebner even got away with the ugly 5...f6 but typically followed up with 6...Bg7, whereas I've done some research (though its been five years or so and I havent run it through the latest engines) showing that depending on where White retreats his bishop, Black can play ...Be7 on move 6 and possibly follow up with ...f5, or ...g5 and ...h5
It was a tough one indeed!