To 'move forward with' something means to progress with it. If you chose to move forward, you agree to make the next steps or to continue in the process. To 'move forward' can mean to advance in position or to progress, or it can mean to propel something or someone else forward. If you move an event forward, it means to make something happen at an earlier time than originally agreed.

1. It was a difficult meeting, but in the end we all agreed that moving forward with the proposed job cuts was the best way to save the business.
2. I asked my client if I could move her appointment forward because I needed to leave work at lunchtime that day.
3. Once we had secured the necessary funding, we could move forward with our expansion plans.

1. I am sad that the relationship is over but I am ready to move forward with my life now.
2. I am happy to move forward with the sale. I have a deposit ready for the car.