It is the responsibility of all players in a team to take the centre passes! Sure, it is the centre that steps into the circle with the ball, but she/he can only do this when everyone else in her/his team is on the right side of the transverse lines. Otherwise the team may be penalised.
When a goal is scored, immediately sprint back to your starting position. You only have to be behind your transverse line. A foot on the line is ok since the line is considered to be part of your playing area. For tactical reasons there are some positions that are better than other and which you would seek depending on the opponents and your own team, but for now focus on speeding up the game.
Check the umpire’s arm to see the direction of the centre pass. It is common that the umpires during a high level game put down their arms when reaching the centre third.
If you are centre, never stop or slow down on your way to the circle. It is not your responsibility to check whether everyone is on the right side of the transverse lines and wait for them. It is your responsibility to restart the game as soon as possible. To stop before reaching the circle and then jump into it should be avoided since this may disrupt the flow of the game.
It is completely ok if the centre is already standing in the centre circle when receiving the ball. Hence no need to step in to the circle and the umpire can signal immediately.
If someone in a team is too slow he/she may be penalised. For example, if the GD is still walking back to the goal third when everyone else is set to restart the game (including the centre) the GD may be penalised for delaying play. Then a free pass to the non-offending team is awarded where the infringement occurred. In a game where goals are being scored in nearly every attack and it is your team’s centre pass, delaying play is similar to giving the other team a free goal. Avoid this!
It could be smart (and fun!) to practice the process of restarting play after a goal has been scored. Time the procedure. See how long it takes. During a high level game it usually doesn’t take more than 3-4 seconds between a goal has been scored and the next centre pass. Sometimes even less. Can your team do it in less than 4 seconds?