When you beat a level on easy, you earn one star. To earn the second and third stars on a level, you have to play it on medium and hard mode, which are unlocked for one specific level by beating the difficulty level before it. Once you beat the hard mode, then you can quick-look the level to earn a rapid reward for it.
Fighting in easy and medium mode is a good old-fashioned button masher. Go in, smash out some combos, and move on. However, when you play a level in hard mode, you'll get killed quickly if you get caught slipping. Enemies will maneuver out of the way of your hits and try to surround you, so forget the combos and play hit-and-run to keep yourself from getting killed.
You have two main methods of upgrading your character. One is to spend coins in order to increase their level, which will increase all of their base stats. The other is to evolve them, which will increase your max level. Collect an item specific to that character, such as hats for Lee, and once you have all of the ones that you need, the evolution will be available.
You can play the game on either landscape mode or portrait mode. Landscape mode is far easier, as it increases the area of play, and thus the area that you can use and still keep the rats on the screen. Portrait mode shrinks the area and adds useless vertical height, making it tougher to play with. The difference is especially pronounced on the taller, narrower iPhone X.
If you're fighting one-on-one with an enemy, move forward into them to grab them, and then tap rapidly to punch them without the threat of them punching you back. If you are fighting against a group of enemies, the grab-and-punch is risky, but grab and then swipe to the side to throw them into each other for extra bonus damage.