The faster you complete the mini-games, the more energy you will be rewarded. Your goal should be to not get anything under about 12 energy per mini-game.
To get your 30 units of energy back instantly, advance the time on your device by 60 minutes. Afterwards, set the time back to its correct value.
You must find objects quickly and create combos by finding objects one after another before the timer runs out in order to score well. If you are a few thousand points from unlocking the next star, you should still attempt a perfect score. Any extra points will be carried towards your next star. The more points you have, the fewer replays you will need in order to earn stars.
When given the choice between different types of investigations, always select ones that have a timer. By doing this, while you are playing to get more stars, the time is advancing and you will not have to wait afterwards with nothing to do.