The game saves automatically . Make sure you completely finish a chapter. Even when it says you complete a mission it doesn't mean the chapter is done. Same with missions on the map. When you can't use your weapon anymore after you finish a mission/chapter it means you finished that mission/chapter. You can also manually save the game by clicking the minus symbol on your switch. This allows you to save your progress.
The map becomes available after the first tower has been activated. So are you sure you didn't skip anything? You might have forgotten something.