What should people like me focus on? I've got some Street Fighter experience but not MUCH.  I've learned how to team-air juggle in the most basic way. I can also bust out basic combos fairly well but not the more advanced ones. The "Mission Mode" is SUPER hard for me.

I'd recommend doing missions actually. Grind out a characters combos that you like, for hours and hours. Once you've beaten that stuff into muscle memory, try out another character. Then go into training mode and worry about controlling space and movement, then worry about incorporating assists in your combos/gameplay.

Seems like good advice. Those missions are hard though :) I'm on a dualshock 3 and that seems to work OK.

Like Ultima said, literally choose a character and just grind out their missions'.  First thing I do when I play every time is play through each one of my main team's missions.  It's come to the point where I can almost do them first time everytime.  Then when you're in a match and have an opportunity to do some serious damage with one of those combos it just kind of happens and you don't even think about it.

I'd just keep playing as much as possible. I'm not great myself but I think the biggest keys to success are always the basics. Things like movement, blocking, spacing and just using techniques you are 100% confortable will get you farther then practicing combos. You'll never get to use those combos in real matches unless you have the base skills to set them up. Start slowly adding in more complicated tactics only after you've got a rock solid base. In Tekken I've heard a good barometer for mastering a character is to play 3000 matches with them. So it's a long slow process of just doing what you know how to do to the best of your ability and slowing adding new thing in small doses. It's a problem I have when I learn new fighting games, I want to do everything right away and I don't play very fluid. It makes me play very flow-charty because I'm thinking too much and my basics go down the toilet.

Actually I'd say learn the basics of when to block, attack, call assist, etc (only way to do this is just to play). Then when you hit the wall of finding opponents that punish harder then you (but pretty equal otherwise), go ahead and start to hone in on those combos. People that train exclusively on combos first will feel so dejected when they are out zoned and such that they are usually the first ones to complain about damage scaling or how being kept out by a runaway/zoning character is ultra cheap just because combos is all they know. 

Wow, all this advice makes me feel like I will eventually hate this game if I should be spending most of my time training in mission mode. Im a beginner too, and the mission mode is hard. My advice is get into some player matches or even jumping head first into some ranked and get a feel for the intensity and the strategies employed by other people online.

oh man mission mode is so tough for me, having said that I persisted on Ryu's 6th (or 7th I can't remember) for almost an hour and finally got it, now I can do it about 60% of the time (gotta love muscle memory :D)

Go into training, play around with basic combos and launchers.  Maybe do some missions, if you're super new to the style then some basic stuff would probably benefit you more, just easy stuff like L > M > H > S > M > M > H > S nearly everyone can do that.
 
Also practice blocking, it's the most important thing you will ever learn.
 
Trying to be too flashy before you're ready will get you a loss, it gets me a loss very often when I choke up on a combo

You can also just play Arcade mode with different characters. Arcade mode is simple enough (I got the "Saving my Quarters" on Normal mode on my second runthrough) that you won't find it too challenging, and you'll learn about each character you play, as well as about each character you face. However, as you may have guessed, the styles and tactics of Arcade enemies aren't similar to real players, so you can't let yourself get used to the AI.
 
Edit: Also, I saw you were using a Dualshock. You may want to grab an arcade stick if you get really into multiple fighting games, or become competitive in one. It's a big help.

The posts in this thread help a lot.  I'm a little (ok a lot) overwhelmed by this game.  Seems like it has legs though, I can see people playing this for years. 

Having combos down into muscle memory as everyone says is the best thing. After that you can pick up opportunities for damage in the first place, where you might have missed them before, and then start being creative. 
 
Although before all that I'd say get around the cast a bit so you know who you'll like first before hamming the combos.