What are some attacking plans for white?

In move 3, black played 3...d6. Is there a better way to exploit that?

Later in the game, I felt forced to play 13. e5 resulting in the trade of many pieces. Was there a way to avoid it? An earlier h3 preventing Ng4 earlier?

Also, my kingside attack seemed to go nowhere. Any alternative plans if I want to try to play tactically? (For the sake of practising)

Thanks!

Attacking? It’s not always about attacking bluntly. Not blundering the b2 would have been an issue. Btw, Bxc6 instead of Nxc6 is a more common approach besides other setups.

Play it more quietly. There are many improvements, center strategy like rooks in the middle first. If you feel like attacking you swap too many pieces ending up overstretched. Too few pieces for controlling two much space, Black takes over.

Yeah, e5 was by no means forced.

You should play the Ruy Lopez like you would milk a cow.
If you want to attack then play King's Gambit, Göring Gambit... anything with Gambit.
3...d6 is a solid but passive variation. You cannot exploit, but you enjoy a space advantage.
6 Nc3 may be better: develop a piece, let him spend a tempo on trading instead of trading yourself.
After you lose the tempo with 7 Bd3 you have no advantage at all i.e. no foundation for any attack.
You should have traded queens hoping to exploit the weakness of the doubled c-pawns in the endgame. As the game went, the open b-file compensated for the weakness of the doubled pawns.

you played it right, except you dont have to give him your queenside. So b3 and/or Rb1 and then continue like in the game. Also, as Sargon noted, you dont need to mate him, put your pieces in active positions so that he has nothing better than exchanging them, the idea is to win the endgame Rubinstein-style:

Actually I thought of this classic endgame as well. But I decided not to extrapolate too far. :D

Here is an example, where white sets up an attack in this variation with some help from black and where pawn b2 is poisoned.

@brianmak #1
I found it rather impressive to see how you kept a sort of initiative going in your game.
However, to become a better chess player, you need to improve on various areas. With only wanting to attack you will likely suffer from time to time against stronger players, or prepared weaker players.
I didnt finish "How to reassess your chess" by IM Jeremy Silman, but I found the introduction chapter to be refreshing and interesting.
It talks about how lots of chess players didn't work on their chess foundation, and get stuck at some level.

@brianmak #1
How did this game end? - Or is it correspondence and still going? - I am having ethical troubles with commenting on a struggle, that should involve 2 players only. Also, my own corr experience lasted only a few years when i was still learning a lot and comps were quite useless. So i am assuming, you too use it as a tool to immerse yourself with chess effort, which i respect very much. Just be prepared to learn a few lessons along the way. :-)

If you're looking to 'attack' then the Ruy Lopez is probably not a good opening for you. Something such as Scotch or Italian would be better. That said 3...d6 is passive by black but not a mistake by any means. There's nothing to 'exploit'.

@SlowSlug it ended in a draw!

Thanks for all your suggestions! In fact I think myself as more positional, but trying to play more tactically. I should try the scotch or other sharper lines instead indeed!

@tpr @Sarg0n thanks for the endgame suggestion! It's impressive!