Keep in mind you can change the controls in the PROFILE section on the main menu (key config). My preferred controls are A (Melee) = Z, B (Weak Bullet) = X, C (Strong Bullet) = C, D (Dash) = Spacebar, A+B (Switch Card) = S, and B+C (Use Card) = D. Of course, this is just my preference though
The cards come in 3 different varieties: grey Skill Cards, Yellow/Orange Spell Cards, and red System Cards. Before I explain them, keep in mind that you draw cards from your "deck" of 20 (pre-assigned at the start of the match) by attacking or blocking; you'll notice that when you do these actions, a blue bar fills up one of the card slots in the bottom left hand corner. This is your "hand", and you can only hold 5 cards in your hand at any one time. When you use a card by pressing B+C, you lose that card and gain its effect (read below).
Skill cards are used to change or level up your current skill set (the attacks performed at any time using the command inputs, like 236 Bullet or 421 Bullet for instance). Each character has a default skill set (at LEVEL 0) with attacks mapped to these command inputs, however if you use a skill card with a different attack (an "Alternate Skill Card"), you can change the attack for that command input. More importantly, if you use a skill card with the SAME attack as what is currently in your repertoire, you can level up that attack for that match, which can lead to more damage, more knockback, ability to use it in the air, etc.
Spell Cards are the easiest to think of: consider them to be "supers" or "ultimate attacks". They have massive and more damaging effects than regular attacks, and smart use of them can secure your victory. The catch to these cards though is that, in order to use them, you have to discard some other cards from your hand at the same time. The better the spell card, the more cards you have to discard in order to use it. In battle, you're only able to use a spell card if it's surrounded by a yellow glow when in your hand, and you'll notice that yellow glow also appears around the cards that will be discarded if you use that card then and there. If there's no yellow glow, you don't have enough cards in your hand to use the spell card.
System cards are universal to all characters, as in all characters are able to use these cards in their decks. They have (usually) one-off effects that can range from interrupting an opponent's blockstring to changing the current weather effect. Some of them work a bit differently (like Sacrificial Doll, which increases your defence by a set percentage for the entire match).
By default, all the characters have their own premade decks, however you can alter them at any time, and have up to four different decks for each character if you so choose. To change or view a deck's contents, go into PROFILE in the main menu and then go to Deck Config and select your profile (if you haven't created your own profile, it'll probably be ProfileP1 or something). Just remember, each deck must contain exactly 20 cards, and only 4 of the same card at maximum.
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strongly recommend using hisouten.koumakan.jp as your library for all the cards though - not only will it give a better explanation of each cards' effects, but it'll also give you a better idea of if it's any good or not, as well as the damage output etc. Plus, as the game screen can be windowed, you can easily tab back and forth between the deck construction screen and the wiki. The premade decks are pretty much fine though.
You can also download a save file that will have all the characters and cards unlocked for each character on the koumakan wiki homepage!!!!
EDIT: Also regarding the card system, whilst it's usually not imperative to level up a lot of the skill cards, keep in mind that the more you have in there the higher the chance you'll draw it earlier in the match. Ergo, if you're playing a character that really NEEDS a skill change, having three duplicates of their alt skill card is a good idea (e.g. Alice Margatroid has a couple of HORRIBLE default skills
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EDIT2: Also, regarding Flight, each character has a unique flight ability as well - some of them are fast flyers with limited curving, some of them are very agile but slow, and Remilia (a.k.a. Metaknight) even Teleports (a.k.a. Dimensional Capes) instead of flying
They all have the same effect of being able to go through projectiles though.
The special moves (the ones that look sorta like "Supers") are done by using the Cards you see at the bottom right/left corners. A+B (card switch) and B+C (card use) inputs can be mapped to a single button as well.
