Four places to focus (in order of maybe importance):
1. SHFFling (mastering each part of the short hop (using x or y), using the cstick or a button for aerials (cstick preferred on many characters), the fast fall and the elusive l cancel (you can use r, l, or z to do it).
2. Teching: Miss a tech and the combos will continue. But don't spam "l" or "r" because you wont be able to tech for 20 frames after a failed tech. Maybe hit up a cpu and focus on it.
3. DI: Directional influence. There's a whole lot to be said on the topic but the basics is when you are hit by any attack, you have the ability to control your trajectory by using the analog stick. If you are being thrown or combo'd, try holding the analog stick towards or away from the opponent to make the next links of the combo harder to connect with. If you are being hit by a kill move, you want to hold the analog stick perpendicular to the kill move's trajectory, as that will bend the trajectory in the most favorable way for your survival.
For example, a move that kills you straight up like fox's usmash should be DI'd left or right (perpendicular). Most kill moves in the game send you up and away from the stage, so the best way to DI a kill move is generally up and towards the stage.
4. Wavedashing. Learn how to do it and of some of its uses on mobility. Its good to dash in a direction and wavedash in the opposite direction (feign a commitment), or to wait for an incoming attack and wavedash back into doing whatever punish you want.
Finally, remember the basics. Shielding is VERY good, doing quick aerials Out of shield is very good, shield grabbing will destroy most amateur approaches (this is where you hold shield and hit a), crouch canceling is good (holding down reduces knockback of attacks by 1/3, allowing you to counter attack).