what you should do is get king and rook far apart and then check king so he cannot hold on to his rook. :)
How do you win a Queen and king vs king and rook endgame?

it is:
1.the trick is to keep checking him as that will cause him to move away from his rook then capture his piece.
2. you could take a draw
3. you could attempt checkmate
but my final idea is that you should ask a GM or an IM.
It is difficult, and even some grandmasters will draw having the Queen if they have not practiced it recently. My suggestion is to play it against a computer with tablebases. Even if it never appears in your games, you will gain a lot of understanding.

In practice it's quite easy as you can usually rely on your opponent playing too passively and/or dropping their rook to a tactic. I wouldn't have a clue how to go about winning it against perfect defence. On the other hand, I've never had it in a serious game. Life is just too short to worry about some things...

Find a fork, skewer, or pin. Check the king until he is forced into a position that will give you one of these.

what you should do is get king and rook far apart and then check king so he cannot hold on to his rook. :)
What type of style are those pieces?

On move 78 black can move the rook to g3, which forces a draw by stalemate or insufficient material.

there's a free chessable course on this. but the idea is basically the attacker will force the defender to the 7th rank and the rook on the 6th after that force it the rook to relinquish the 6th rank to the 7th rank while the forcing the king to the backrank. then either fork the king and rook or if rook doesn't want to leave the king force it into philidor position and eventually the rook will leave the king then you'll be able to fork the king and rook.
How does one get checkmate to ensue if you are presented with such an endgame. I heard it was a difficult win with best play but is there a certain method that should be followed like with the other endgames or is it just an endgame that has to be figured out.