Wall of text incoming-- I'm not the best at explaining things, but I did my best! Hope it explains some things and helps everyone understand how to do longer air combos and stay aggressive all the time.
Default buttons and controls:
A - Your standard attack button. Does different things depending on the weapon you have drawn currently, pretty straightforward. Draw your shape and press this to make it appear. Not much to say.
X - AKA the homie throw. This move seems kind of whatever at first, but it's vital for good movement and control during fights. This move is important for two big reasons:
1. If you keep using this against an enemy, a red circle will appear them. This is kind of like a lock-on of sorts. Basically the next attack aimed in the direction of that enemy will teleport you straight to them with your attack. This is great for quickly maneuvering around a battle and it's also very good if you're fighting a single large enemy and are trying to weave around them.
2. If you continue to hit them with it, they will eventually be stunned. I think they can still attack while in this state, but they cannot move. This stun allows you to launch the enemy guaranteed so if you want to do a combo, you should start with this.
Also refills your battery gauge*
Y - Makes you dash. This buttons seems pretty pointless in combat, but it actually functions a lot like the blade mode cancel in Metal Gear Rising. Basically during any action you perform, you can tap this button to completely cancel out of it (I'll just refer to it as a dash cancel). So at any point during any attack you can hit this to completely recover to your default stance (no weapon out). This is useful for keeping up constant offense as well as making any attacks safe.
ZL - Jello block. Not much to say about it, it blocks and reflects attacks. The window for this is very lenient, so you don't have to worry too much. Can dash cancel to recover faster.
ZR - Spring dodge. Uses 1 battery icon in Story (apparently challenge mode in the demo is set to Easy stage)*. Basically the dodge from Bayonetta. This game has the dodge offset system that Bayonetta such a brilliant game, so if you're familiar with it you'll feel at home. Not going to explain it as it's kinda hard to put into words, but I'm sure there are guides on it.
Wonder Rising is the default launcher for every weapon is B then A (jump then attack in quick succession). You will rise into the air with your weapon allowing you to aircombo.
I'm not going to go too into the Unite Morph system, as it's been explained a million times. The most important thing for combos however, is the fact that you can actually draw a weapon's shape and hit X instead of A to have the weapon come out and do a preset combo or attack and disappear again (I'll refer to these as assists from now on). It's important to note that you can only do this as long as you have spare heroes left over. So this means if you draw a sword using all of your heroes, you cannot use any assists. Because of this, I prefer to keep my weapons small or medium sized. I'd rather have more options over raw power, personally.
The assists always do the same thing:
Red - Standard AAAA combo
Blue - Standard AAAAA combo
Green - Shoots 360 degrees around you
Pink - AAA combo
Yellow - Single strike
I'm not really crazy about Pink or Yellow's assists personally, but I'm sure they have their uses. Red and Blue are great for combo filler and general assists while fighting. I like Red for air combos and Blue for crowd control on the ground. Green is a great crowd control tool since it shoots all around you.
R-Stick vs. Touchscreen - Generally, both schemes work fine for drawing. I know some people prefer the screen but I like the right stick more. You don't have to move your thumb too far from the buttons (especially the vital dash button) and you really don't need to be THAT precise with the shapes in the demo besides Yellow. It's also worth pointing out that the game triggers slo-mo when you begin drawing, but ONLY in the story mode. I don't know if the final game will change things (I'm going to assume the final difficulty will not have it) so because of this I recommend the stick. It's faster to get to and gets the job done. If you can draw your shapes during combos using the touchscreen, more power to you.
General combo stuff:
Use the X button a lot! I like to harass enemies with it first since it can cause the stun state which makes air combos guaranteed. Sometimes you can launch unstunned enemies, but it's too random for me. I like guaranteed launches. The teleport feature is also VERY good as you can tag an enemy and go focus on another one farther away and then teleport back to it after you're done. It's also great against big targets since you can basically fly around them attacking and they'll be mostly helpless.
This game (at least the demo) doesn't really have many strings. Each weapon has one string on the ground and one in the air. Adding pauses between attacks doesn't seem to change anything. It's pretty clear there will be unique moves in the final game though. But that's where the dash cancel comes in!
The Dash Cancel is pretty much the most important thing to combos and general offense in this game. At any point while attacking or blocking you can hit Y to cancel out of it completely. It recovers you to your default stance immediately allowing you to follow up with whatever you want. So this means with Blue equipped you can do something like this to a stunned enemy:
AA, wonder rising, jump.AA, dash cancel, j.AA, wonder rising, j.AA, dash cancel, jAAAAA then dash cancel away to continue your assault.
Also note that you just have to tap Y, you don't have to hold it or move in any directions or anything. A simple tap works.
I previously said that Red and Blue are great combo fillers, so what you're going to want to do is start mixing their assists into your air combos and ground game. So starting with Blue you can do AA, wonder rising, wonder rising, j.AA, dash cancel, j.AA, summon Red assist, j.AAAAA. If done right you'll basically be juggling the enemy all the way into the upper edge of the screen (I've had a few instances where they're no longer even visible). You can also swap your weapon mid combo to get even fancier so in theory it's probably possible to do something like: wonder rising with Blue, dash cancelled after one hit then swap to Green for his wonder rising then his mash follow-up while calling the Yellow assist to slam the enemy back down.
So all in all, while this game does not have the combat depth or variety of DMC3 or something, there are some options to get creative with the tools here. So while the game may seem mashy at first, it gives you the tools to do some more elaborate combos. I didn't do exact tests of the combos listed, but in theory they should work fine (or at least with some small tweaks). I think for the most part, combo monsters will enjoy the playground this game gives you over Metal Gear Rising due to all of the tools you have here.
*=Thanks KniveSmith!