Many moons ago, I was a teenager who thought he was the best of the best when it came to certain fighting games. I had a wake-up call when I entered my first real tournament and saw how high-level the top players were. Shortly afterwards I picked up the Japanese version of Super Smash Bros. 64. My friends and I played the game almost daily, well before the U.S. version released. We played with all items on and always in four-player free-for-alls. It was some of the best times we had as gamers, and we still reminisce about it regularly.
Playing Smash Bros. casually is how Nintendo always intended the game to be played. In fact, it was only recently that Nintendo embraced the competitive scene. Changes were made between Melee and Brawl to actually stifle growth of the competitive scene (random tripping), but despite these efforts, the competitive scene for Super Smash Bros. is one of the largest gaming communities in the world. If you're a casual fan of the series, do yourself a favor and check out the competitive scene. It may not be your thing, but you're missing out if you don't at least try to experience it.
If you're a casual player, you may not know how to get into the competitive scene. Some people think there couldn't possibly be a Smash community where I live. First and foremost, you'd be surprised at how many cities in the U.S. and around the world have a thriving Smash Bros. competitive community. The first place to start your journey into the world of competitive Smash is at the Smash Boards forums. In the Smash Tournaments section under Regional Zones, there are hundreds of threads for a plethora of cities across the world. Most of the people are friendly and more than willing to offer invites to local gatherings.
Once you've greeted the community and possibly made your way to a local gathering, it's time to figure out where you stand. You have two options here. It's virtually guaranteed that you're not good at Smash Bros. compared to the long-standing competitive players. Don't worry, that's normal. In most cases you're only as good as your competition, and even if you beat up on all the neighborhood kids, there's almost always someone out there who's better than you.
You have to decide if you want to improve your skills and challenge the better players in your community, or if you're content with your current skill level and only wish to watch high-level Smash. Even if watching top players go at it isn't fun for you, at the very least, you should spectate at least one tournament. If it's local or within your budget to travel, go in-person. If you can't travel to an event, watch them on Twitch.tv. You can find upcoming tournaments on the Smash Board forums, which almost always have a link to the live stream enabling you to watch from the comfort of your home. Of course, there's nothing quite like the hype and hysteria of being at a tournament in-person, especially during the finals.
If you decide that you want to improve your game, you first need to understand your current level of play. If you beat up on your friends all the time, imagine what would happen if you had to fight yourself. What would you do to counter all of your strategies? Now think about what you would do to counter those counter strategies. This is the line of thinking you need to take to improving your Smash Bros. skills.
Getting into the proper mindset is the first step. You can't get discouraged when you lose because chances are it will take A LOT of losses before you even begin to see significant improvement. However, when you do lose, ask the opponent why you lost. More than likely, the opponent will be happy to give you some advice. You're only as good as the competition, which means the more you play against better players, the faster you'll improve.
Smash Bros. is a game that's big on movement, spacing and advanced techniques. While it's important to play against other people, it's equally important to practice at home on your own. You need to have solid execution to improve your skills, and the easier it is for you to perform the combos and techniques you want (when you want), the faster you'll be able to improve. In many cases, this boils down to simple muscle memory. The more you perform a specific motion, the easier it is to execute and the more likely you are to perform it perfectly under pressure.
Once you have your execution down and you've played better players as often as possible, the next step is to play in as many tournaments as you can. It's a virtual guarantee that you will not perform well in your first tournament. This may not be because the other players are better. Rather, it's a common fact that most people do not perform well under pressure or in unfamiliar surroundings. You're accustomed to playing Smash in your living room or at local gatherings. When it's time to play an out of state tournament, it's a completely new venue and unfamiliar surroundings. The more you get accustomed to playing in unfamiliar situations, the more comfortable you'll be.
As a final note, it takes time and patience to reach a level where you can compete with other high-level players. Your skills won't quickly improve, but if you give up or slack off on practicing, it will take even longer. Stick with it, have fun playing and you will improve over time.