Is there a way to start one game on the iPhone and continue it going on to your iPad?
This is dependent on how the game's developer has coded it. In principle it is possible that the developer saves the state somewhere on the internet but my guess is this is only normally done for games with a network or social component. It may even be possible to do a sync via iTunes on a desktop computer shared by the iPhone and iPad (i.e. sync the iPhone with the computer then plug the iPad into the computer). However, all of this depends on how the game was coded and as far as I know there is no general way of doing this.