[fen ""] [Result "1-0"] [ICCause "2"] [ICEcause "4"] 1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 Qe7 5.Qxe4 Nc6 6.d4 b6 7.d5 Nxe5 8.Bf4 Nf3+ 9.gxf3 Bb7 10.Nc3 f5 11.d6 cxd6 12.Qxe7+ Bxe7 13.O-O-O O-O 14.Bc4+ Kh8 15.Rhg1 g6 16.Nd5 Bh4 17.Rge1 Bxf2 18.Re7 Bxd5 19.Bxd5 Rad8 20.Bxd6 Bh4 21.Re5 Bg5+ 22.Kb1 Bf4 23.Bxf8 Bxe5 24.Re1 Bxh2 25.Be7 Rc8 26.Bf6# 1-0
There is not much of an endgame here. Black played the opening poorly, blundered a piece and white only had to convert the position with an extra piece which he did reasonably well.
Generally, in a clearly winning position, it is a bit meaningless to ask for better ways to play, or at least you need to specify what you mean by "better".
If you want the objectively best play you should have a computer engine analyze your game. This will tell you the best continuations, which is very roughly speaking, the shortest way to mate for the winning side or the longest way to mate for the losing side.
However as a human player that is not what you usually want in such situations. If you are on the losing side, you are basically hoping for mistakes from your opponent and you do everything to help him doing that (see also this recent question and answers), even if that means playing an objectively worse move. As such, in a lost position you:
- avoid simplification of the position, i.e. avoid the exchange of pieces
- try to complicate the position for instance by having more than one center of action
- try to set tactical traps
- try to create imbalances which are often more difficult to play (e.g. being a piece up is often an easier win than being an exchange plus a pawn up)
- go for a direct mating attack
- play quickly, hoping to get your opponent in time trouble
- aim for getting stalemated (only works rarely and only in endgames)
Conversely, if you have a winning position with material advantage (white here) you often:
- aim to simplify the position, by exchanging pieces
- avoid complications, e.g. by keeping the position closed to some extent
- keep your king safe
- slowly convert your advantage without rushing it
Regarding your game. The first main blunder by black is 6...b6. At this point black needs to play immediately 6...d6 in order to get the sacrificed piece back, taking advantage of the pinned knight on e5 while the white queen is unprotected. 6...b6 allows white to just defend his queen after which he is a piece up. Even worse, with 6...b6 black self-pins the knight on c6 and white could reply 7. Nc3, protecting the queen (thus winning a piece as above) and in addition creating the annoying threat Nd5 which attacks the black queen, threatens Nxc7+ and if the black queen leaves the e file you have deadly discovered checks with the knight on e5.
With 7. d5 white gives away all the advantage, losing his extra piece.
With 8. ...Nf3 black returns the favour. Most likely he was hoping to win the white queen, but did not take into account that the queen is protected after 9. gxf3. Now we have a clearly winning position for white and if white just makes normal moves, develops pieces, exchanges pieces, protects his pawns, there is not much that black can hope for. Creating complications after the queens get exchanged will be difficult because there is little material left and white does not have any major weaknesses. In what follows white made no major mistakes, but from a practical point of view, I would have played it more calmly/less tactical. Specifically:
- d6!? This opens the diagonal for the bishop on b7. Why not just exchange queens, castle long, develop the bishop from f1, put the rooks on the open e-file and play it safe. There is not much active that black can do.
- 0-0-0?! This allows 13. ... Bxf3 where black wins an exchange. So essentially white changes from playing with an extra piece to playing two pieces against a rook which is still winning but much more difficult to convert. Perhaps white should have left the king in the center on d2, where it is difficult to attack. The weak pawn on f3 could be protected with the bishop or sacrificed if you are going for active play.
- ... 0-0? Black's best chance was to go for the Bxf3 line mentioned above.
- Rhg1 Not a mistake, but 15. Bd5 would exchange black's strongest pieces and after 15...Bxd5 16. Nxd5, white would end up with a very strong knight and black with a very weak bishop on e7, which could be exchanged if needed, further limiting any potential counterplay.
15...g6 Still black could have taken on f3.
- Rge1 natural enough move to occupy the open file, but Bxd6 could be a very good alternative, winning a pawn and threatening Be5+ with (almost) mate
- ...Rad8 Too passive. If black hopes to get anything out of the game he should at least occupy the open e file by playing Rae8.
- Bxd6. Here Bh6 would win an exchange because of the threat Bg7++.
- Re5 a strange square for a rook. The discovered attack on the f8 rook is not dangerous and if you wanted to do that you could as well have done this with 21. Re2 which avoids any attacks on the rook as in the game. Generally, if the opponent has only one bishop, it is often a good idea to put your pieces on squares of opposite color to those on which that bishop operates. There is nothing wrong with 21. Re5, but from a practical standpoint it unnecessarily complicates the position.
- Re1 As above, nothing wrong with this move, but it somewhat complicates the position. You have to calculate all kind of intermediate moves (like Bxb2, Bg3....). Easier would be to just move away the attacked bishop with 24. Bh6