Reach for the rook on h1 first and you have just played a rook move. That is not an arbiter being awkward, it is Article 4.4 of the FIDE Laws, and in a rated game your opponent will hold you to it. So the order is king from e1 to g1, then rook from h1 to f1, and the same hand does both. The Laws never actually print the words “king first”, but that is what they add up to.
Touch the rook first and you have played a rook move
Here is where the rule bites. Article 4.4 of the FIDE Laws covers castling and touch-move together. Touch your king and a rook, in that order, and you must castle with that rook if castling is legal. Deliberately touch a rook and then the king, and you may not castle with that rook on this move at all; what happens instead is governed by ordinary touch-move, so you have to move the rook you touched.
Think about what that costs. You meant to play O-O, you grabbed the h1 rook out of habit, and now you are obliged to play something like Rg1 or Rf1, which does not just fail to castle, it kills the castle for the rest of the game because the rook has moved. One careless hand, and the king is stuck on e1.
Nobody enforces this in a park game. In a rated tournament, with an arbiter in the room, your opponent absolutely will.
One hand
Article 4.1 is one sentence long: each move must be made with one hand only. That applies to every move, and castling is the one where players break it without noticing, because two pieces are travelling and it feels natural to use both hands.
It is also the one players get away with, because a two-handed castle in a time scramble is over before anybody can object. Fix the habit while you are playing at home. King first, one hand, rook second. It takes a week and then you never think about it again.
There is a related move worth knowing about at the same time: if a piece is sitting crooked on its square, say “j’adoube” before you touch it, and the touch does not commit you to anything. Say it before, never after.
The point of no return
The move is not finished the moment the king lands. Under Art. 4.7, castling is complete when your hand releases the rook on the square the king crossed. But you cannot change your mind either, because the same article says that once you have released the king on its new square you no longer have the right to play anything except castling on that side, if that is legal. Put the king down on a square where castling is not legal and you must make a king move instead.
So the decision closes when the king touches down on g1 or c1, and the move ends when the rook touches down on f1 or d1. Between those two instants you are committed but not finished. It is a distinction that matters about once in a career, and a lot on that day.
Get the order into your fingers on a real board rather than a screen, where you drag one piece and the software does the rest. Set up a position, castle both ways twenty times with one hand, and then go and solve a few castling puzzles so the move is in your head as well as your hand.