To progress in any given Case, you must complete the tasks listed in the "Tablet" view. To access this task list, click the "Tablet" button in the bottom left hand corner once you've opened a Case.
You will require Stars to complete certain tasks (such as talking to suspects or playing evidence mini-games). You can earn Stars by playing Crime Scenes.
There are other tasks that merely require some time to complete, such as autopsies or lab analysis. While you are waiting, you can continue investigating Crime Scenes in order to earn more Stars. If you don't want to wait, you always have the option to purchase an Instant Analysis.
As you complete the aforementioned tasks, more Crime Scenes will unlock and you'll move along further in your investigation. If you ever feel you are blocked in a Case, please click on your Tablet view to see if you have any outstanding tasks to complete.