Don't be afraid to click on the '?' near the top left side of the screen, if you need a hint. A total of three are available in each section.
If an item is usable, tapping it twice will allow you to examine it more closely.
Inventory items sometimes must be placed on other objects to solve puzzles. Note that an item changes from transparent to opaque when it is able to interact in such a manner.
Tap the eyeglass icon, located near the screen's right side, and you'll be able to examine objects in the environment that you might otherwise miss.
When you're out of hints or just don't want to use them, tapping and swiping everything can go a long way. You should also try the various items in your inventory and the eyeglass, just to make sure you have truly exhausted every possibility.
You can swipe the screen to adjust the camera angle, which may be helpful if you're not noticing something crucial to the puzzle at hand.