Yes - you can transfer both GTA 5 story mode and GTA Online progress. For story mode, you need to upload a save to the Rockstar Social Club, from where it will be imported. Your Online character can then be imported in-game. Note that only one save game can be stored in the Social Club at any given time.
Players have documented waiting periods when importing saves - it isn't instantaneous, and that is normal. A screen will inform you that the importation is still under way before booting you to the main menu. After the process is complete, you can pick up where you left off in story mode. In Online, you can choose to start fresh with the Career Builder, or continue from where you were.