Play the solo modes is a great place to start, that a lot of people seem to over look. If you are trying to grasp fundamentals, character properties and move sets, and practice things like timing and spacing, the different Solo modes will definitely help. Smash Run will give you a great place to practice interacting with the physics of the game and how to anticipate against opponents moves (most of the baddies have a decent wind up). Classic mode will give you a way to prep for more normal Arena based combat you'll find in a tourney (sans the items), while All Star will focus your mind on how to improve your defense. (Fighting an endurance match forces you to weigh your options carefully to reduce how much damage you'll take, great for when you need to play defensively) Target Blast and Multi Man Smash will help improve your accuracy (Blast for knockback angles, Smash for hitting moving targets), and Home Run contest will improve your combo knowledge. You won't bore yourself running drills like this is basketball camp using these modes.
Once you feel comfortable with a character (or characters. Finding a main can be quite the spiritual experience for some.), then start using the base online modes, and local friendlies if possible, to get a feel for human players. For Fun and For Glory are a good starting point that requires no hoops to jump through, and you can start practicing your reads on a human opponent. This is when the Replay feature and training mode will become really important, as you can record your loses for later (taking a break and watching as a spectator will help you see things you didn't know you were doing wrong) and practice speeding up some of your favored combos in the lab with your replay experience will help you out.
Once you've got a basic handle on reading opponents, you'll want better competition so you can start improving your Match Up knowledge and your optimization. Local and online tournaments, Anther's Ladder, friendlies with a fellow smashboarder, and the previously mentioned online modes are all good ways to continue to improve your mind. Muscle memory will only get your so far if you don't know when to flex them. (This is the stage I'm at, the constant state to improve my knowledge of my opponents so I can make the best decisions at the right time!)
Word of advice, online mode can be hit or miss with lag due to how fighting games connect opponents. Once you get to a certain point, local fighting will always improve you faster than online fights. (Reliable timing means consistent muscle memory with less mental focus, allowing your attention to be squarely on your opponent where it belongs.)