I agree with what others have said. Pick a character you're interested in and just mess around with their strings and specials in practice mode. Try to piece together some simple combos, and then look on their forum to learn their optimized combos (which will be much easier if you experimented first). Do the combos over and over in practice mode until they become second nature, since you'll need to be able to do them without really thinking about it. It really just comes down to repetition.
Try to learn a few different combos for different circumstances. For instance, you'll want a good "punisher combo", which may not deal the most damage, but it can be done off of a quick move that you can used to punish blocked specials or whiffed normals. This is probably the most important combo you can learn.
Next find your character's most dangerous tool (either a normal, a string, or a special) and learn how to combo off of that. For instance, Sonya's Cartwheel special is very good and it launches, so every Sonya player should have a go-to combo ready after a connected Cartwheel. Jax has his quick, safe, low-hitting f4, so learning to combo after that normal or its strings (f41, f413) is imperative to playing that character. (I might be saying the wrong numbers, can't remember, but you get the idea).
Once you've got your "punisher combo" and "offense combo" down, it's time to start working on your "neutral game", which is arguably the most important aspect in fighting games. This is where you and your opponent are both trying to open each other up while trying not to get opened up yourself. There's a lot to developing a good neutral game. You'll need to learn your character's frame data, and it's pretty imperative at a high level that you know your opponent's frame data as well, which can be a grueling task. You'll need to learn how to "poke" with safe normals in order to bait out unsafe or whiffed moves from your opponent, allowing you to punish with a combo. You'll need to learn when it's safe to use certain strings and specials. Finally (and this just comes from playing the game) you'll need to be able to start "reading" your opponent, where you can reasonably predict what they're going to do and react accordingly.
And that's basically a good starting point. After that it's just a matter of playing the game a shit load and getting a feel for how certain matchups play out. Don't be afraid to lose. The first few weeks/months of play should be experimental, meaning you shouldn't be afraid to experiment with different strategies to see what works. This means losing a LOT of matches, but the experience you gain will be invaluable. Basically, don't play to win, play to learn. The winning will come automatically.
Oh yeah, and like @
Rickyraws said, you'll want to watch youtube videos of high-level players using your character.
