To be good at the game, you must first understand its mechanics.
Damage and Knockback
Unlike conventional fighting games, Smash Bros. is about racking up damage, then sending the opponent out of the stage. The more damage you rack, the further they are launched from the stage. If they are knocked beyond a blastline or deathzone, they lose one stock.
Stale Move Negation
To prevent players from only using a powerful attack over and over, the developers inserted a special mechanic called stale move negation. There is an internal counter that records the last ten moves a character used. If there is more than one use of the same move in the counter, that move gets weaker. A good example is a Mario player using Side Smash over and over again. Even though he racked up a lot of damage, his Side Smash will be stale, thus rendering him unable to KO with it. So, it is important to save powerful moves for KOing and use damage rackers early on. If you do use a move too many times, use other moves to "refresh" the counter, preferably weak moves. This mechanic is not bad however, as staled moves actually combo better!
There are some exceptions to this. Some moves are not affected by this, such as Samus's aerial Grapple Beam. Some modes, such as Subspace Emissary, Boss Battles, and Training Mode, are also unaffected by this mechanic.
Sometimes, two moves might hit at once. The game decides which move will hit with a mechanic called priority. Characters with high priority in their moves generally fare better than characters with low priority moves.
Most moves posess this kind of priority. Normal Priority is damage dependent. If the difference between the damages of the two moves is less than 8%, they will cancel each other out. For example, Mario's Side Smash (18%) collides with Kirby's Side Smash (16%), so the moves cancel each other out. But if Mario's Side Smash (18%) collides with Ganondorf's Warlock Punch (32%), Ganondorf hits Mario while Mario's Side Smash is cancelled out. This is because Ganondorf's Warlock Punch does more than 14% damage than Mario's Side Smash does, therefore having more priority. Grabs have priority over all attacks.
Aerial priority is dependent on speed. If Mario and Luigi's Neutral Air Attacks collide, the faster one will cancel out the slower one. If Mario's and Luigi's neutral air attacks hit each other at the exact same time, both characters will get hit. Aerial attacks cannot interact with ground attacks.
This is a special kind of priority as it will not be cancelled out. If a move collides with one with trancendescent priority, they will not be cancelled out. Examples of this is: Falco's Blaster, Sheik's needles and Meta Knight's sword attacks.
Some attacks will look like they have electrical or darkness elements imbued on them. This are elemental attacks. These attacks is the same as any other attack, except they get powered up by certain stickers. Generally only used in the Subspace Emissary.
This is a completely random mechanic that occurs when you start a dash. You will trip and become vulnerable. This mechanic is completely useless and can get annoying at times. Just make sure to be prepared for it.