When you pick up the cherry power-up, your character will clone themselves. Each subsequent power-up will clone them again, and which point synchronizing your clones can get really tough. Learn the art of organizing them, running into corners or walls to "collect" all of your clones into a straight line. Then act like you're only controlling one character.
If you're playing single player, go with Peach every time. Her float ability allows her to accomplish some extremely tricky jumps, and it's a lot easier to get stars. She is slower, but if you master the dash move (simply moving in a straight line for a few seconds), it'll make up for it. If you're looking for a tougher playthrough, pick Luigi.
The Cat Suit is one of the most versatile power-ups to grace any Mario game in recent memory. It has multiple uses, one of which is the swipe delay attack in the air. By swiping, you can stop your jump for a half second, which is enough time to error correct for a swinging platform or a deadly drop.
Additionally, if you fall over a wall with the Cat Suit -- don't panic or jump. Just attach yourself to the wall and climb up calmly. Wailing about will just drop you off the edge. The aforementioned Peach mechanics work in tandem with the Cat Suit to make for some really technical gameplay, so experiment with that combo if you need help grabbing stars.
This is a classic tip for the SNES Super Mario World, but it applies here: if you're on a tough level, go back and "stock" up a few suits. You can press the pause button and go back to any map instantly, and World 1-1 can be finished quite easily with two Cat Suits in tow. You will need to finish the level to keep your power-ups.
If you see any stone slabs on the world map, walk up to them and press A. You may uncover a hidden stage or pipe.