Basic Info
You choose 2 riders, and scan them in. Your HP is determined by both riders stats (base Attack, Defense, and Hissatsu are determined using the front rider's base stats).
Then a roulette chooses 1 of each riders 6 attacks, and compares them to the enemies roulettes. Higher number wins, attacks are applied with strength and defense.
Once per match, you can scan a support card.
All the time you battle, you earn points towards a hissatsu attack, which you can activate at 100% or higher. At 200% you can use both riders hissatsu attacks combined together.
Red number is the hissatsu, usually exclusive to each card.
コウゲキ is attack ボウギョ is defense ダイリュク is health
Stuff below it is addition powers/effects.
Each card has a type in a rock-paper-scissors triangle:
力 = strength which beats 技 = technique which beats 速 = speed which begets strength.
Each gains a minor power boost over the other.
The color of the type symbol is what you want during the color roulette at the beginning, and gives you a power boost. (Red - Strength, Blue - Speed, Green - Technique)
Rear Guard System
Ever wondered why sometimes you (or the enemy) did those little finishing strikes during regular attacks or those special defenses during the most (in)appropriate times? It's the rear guard support system. You see, the finishing strike (オイウチコウゲキ) or support defense (ミガワリボウギョ） is, unfortunately, activated by chance. However, there are factors that decide the probability AND the damage/defense bonus. If you see the pict (or, in the game, under the Att/Def/Life status during the card status screen), you'll see the コウエイ row. Basically, it defines how said Rider will play the support. There are five possible support type:
ガンガンセメロ (Attack relentlessly) : Very very high chance of finishing strike (+70% damage bonus), very low chance of support defense (-10% damage received)
アイテヲ ブットバセ (Knock away the enemy) : very high chance to attack (+60%), low chance to defend (-30%)
サポートハ マカセロ (Leave the support to him) : high chance of attack (+30%), high chance to defend (-50%)
アイテヲ ヨクミロ (Keep an eye on the enemy) : low chance to attack (+40%), very high chance to defend (-90%)
ガードヲ カタメロ (Fortify guard) : very low chance to attack (+50%), very high chance to defend (-80%)
Needless to say, a 80/90% damage reduction is a bliss (or a pain in the ass if it's the enemy) when facing a Rider Kick in the face. The +70% damage is not really useful if you're not really lucky with the slots, but it helps during grunt fights.
Kamen Ride & Memory Change System
Decade has the Kamen Ride system implemented, which, essentially, when you pick Decade for the front, the game then prompts you to pick another (main, basic form) Heisei Rider to transform to, Decade will then transform to that Rider for the whole battle, and his special attack replaced with the transformed rider's FFR->FFA. I don't know if he is treated as the transformed Rider, but FFR cards (which requires Decade to work) work when Kamen Ride-ing.
W has the Memory Change system. In the middle of battle, you can change memory the same way you scan for Special Card (the green button). Memory Change does not count towards your 1-time SP card scanning. Unfortunately (I don't know about the story mode, though), W and his 9 forms are available from the 6th pack.