Navigation and Combat Controls and Explaination

Normal Exploration

Analog Stick: Basic Movement

D-Pad: Adjust The Camera with it.

Face Buttons

Triangle: Allows you to talk to NPCs in town without having to double tap Circle.

Circle: Allows you to lock on to targets both npcs and monsters. Using the D-Pad when locked on to a target allows you to change targets.

X: Cancels last command/breaks target lock/Cancel. Hold it longer will make Kirito kneel(sit) on the floor. Hold it longer near a chair will make Kirito and his partner(if there is available chair) sit on it.

Square: Makes Kirito do a fist pump and say something. Not really sure what it does honestly

Start: Opens the Menu

Select: Nothing.

Triggers:

L Trigger: Resets the camera to face behind Kirito

R Trigger: Congrats your partner. Don't do this unless you've just won a battle or you'll get a ? in response.

L + R: Opens up the shortcut menu. From there you can:

Press L or R to open up the Sword Skills and Battle Skills palette respectively. You can then highlight with the square button to rearrange the skills positions to your liking.
Press the circle button to open the Items menu.
Press the triangle button for the full map.
Press the square button to open up the memo.
Use the up and down on the D-pad to scroll through the log.
And left and right on the D-pad to zoom out and in.

Combat

Analog Stick: Allows you to move around during combat.

D-Pad: Used in conjuction with Circle and L Trigger in order to select battle and sword skill commands.

Face Buttons.

Triangle: Uses the mapped skill from your battle skills, can be changed in the shortcut menu.

Circle: Opens up Battle Skills Palette during combat. Double Tap it to use what's displayed as the icon for it.

Square: Uses the mapped skill from your battle skills, can be changed in the shortcut menu.

X: Cancels out the Battle Skills Menu also Cancels out of the swords skills menu

Select: Changes between Active Effects window and MiniMap.

Triggers.

L Trigger: Opens up the Sword Skills List, used in conjunction with circle and the D-Pad to select skills. The left side of help line down the bottom will display the Sword Skill name and it's effects.

R Trigger: Opens up the partner Commands, used in conjunction with circle and the D-Pad to issue one to your partner. Also can double tap R to congratulate your partner. If the line just states "---" then you'll get a "?" response, else your partner does something and then all members in your party receive 50SP. What you can say (in order) is:

Switch!
Use skills!
Stop that (enemy) technique!
Surround the enemy!
Take care of the target!
--Next page--
Recover your health!
Attack together!
Get behind me!
Defend yourself!

Note that the help line down at the bottom of the screen provides a very rough translation in case you forget what it does. And what it actually does in Japanese.

You can access the shortcut menu anytime, so it's a good way to open the item menu quickly when you need to during battle. Or pin it to your Battle Skills Palette (explained in Skills section).

How to Target Monsters

In the field you have to get close to a monster and have it in your field of vision then press Circle from there you can either press circle again (Rush up and attack it) Or press triangle to throw rocks at it until it comes closer to you (Probably a good way to lure it away from it's buddies, or bring them closer to other players so you can train them) Then press Circle when it's close enough. For multiple monsters, you can press X once and then use the D-pad to change your target after that press Circle to lock on to the target again. Pressing X twice will make Kirito sheath his weapon, use this to escape from the monsters since you can't run while locking on to the enemy.

Skills

After the fight with Heathcliff you start with no sword skills. In order to get some of them back you'll have to hit start and go to skill in the menu. Of course if you chose something other than sword or dual swords then you have to fight monsters for a while until you level up the first level of your chosen weapon. Both Dual swords and 1 handed sword have up to level two unlocked which means you'll be able to get 4 sword skills back if you're using those styles. (You have to spend skill points on the skills though)

Equipping Skills

Hit start and select "SKILL" to access the skill tree. You can check your weapon mastery on the left side. Skill levels (or package) require a certain amount of mastery before it can be opened, and need to be opened before you can buy the skills and equip them, provided that you have enough skill points to both open and buy them. Skills that are equipped will have an orange outline on the symbol to the let of the skill name. You have a limit as to how many skills you can equip at one time. Only one Passive Skill can be equipped at one time.

You can also preview what's in a skill package if you want to check out what's in there, or don't want to use your skill points just yet.

Editing Skills

In the shortcut menu, L and R will open the Sword Skills and Battle skills respectively. During battle, it's L and circle.

Aside from using them in battle, reordering them, you can also 'edit' skills by pressing Triangle. There are 6 slots followed by 2 lines. They are:

Skill
Back
Waist
Ring
Ring
Charm
"Revert changes after skill" option
"Confirm Changes" button

In the Skill slot, you can select which skill it will execute. It will give you a list of all the skills you can equip for that slot. The Battle Skills section will also include battle items like STR Potions in your choices.

For the rest of the slots, you can choose what to equip when the skill is executing ("はずす" means to remove, so you won't equip anything for that part if it's selected)(Not sure if it includes the charge time). This is for adding a bit (or a lot) more edge when executing the skill. Or changing your equipment to fit the situation more.

Pressing Triangle will blank it. This means removing the skill (if done at the Skill slot) or no changes (if done at the equipment slot).

Selecting "ON" at the "Revert changes after skill" option will change your equipment to how it was before you executed the skill. Selecting "OFF" will leave the changes as they are. Note that if all the equipment slots are blank, this option will be greyed out.

Skill Points and Mastery

You need to equip the corresponding weapon type to raise the skill mastery, and for ten Skill Mastery points you will gain one Skill Point. You gain Skill Mastery Points by attacking the enemy and is based on your number of hits that you dealt to the enemy. The number of skill mastery points that you get depend on the difference between your level and the skill mastery points that you already obtained for that particular weapon type. So, you'll gain Skill Mastery Points at a faster rate if you still have little mastery points in a particular weapon type but your own level already quite high. Enemy's level also played a factor in gaining Mastery points. You can gain a lot more Mastery points by fighting high level enemy on higher floor. And it looks like you can gain skill mastery points faster too, if you haven't opened up as many skill packages in weapon category that you're using (needs confirmation). Skill Points can be used anywhere in the tree. Skill points might be capped at 99 points, so use a lot at the start!

Using Sword Skills

In battle, there's more to using Battle Skills than just executing it. When both you and your partner execute a skill at the same time, Kirito's and his corresponding battle partner's faces will cut in for a brief moment. Deals additional damage, provided that the enemy survives the attacks.

Landing Sword Skills one after another will result in a 'Skill Chain' and will deal additional damage depending on how long the chain goes for. Needless to say, the longer the chain, the more additional damage it will deal. You cannot execute Skill Chains by yourself. Make good use of that "Use Skills!" partner command in battle.

Switching is also one way to continuously execute Sword Skills. It also reduces risk for the person switching out. To do this, select the "Switch!" partner command while you are charging or whilst executing the skill. For skills that move to another screen when it executes (e.g. the level 5 Dual Sword skill, Star Burst Stream), you must command them before it's executed. You will know when your partner is preparing to switch when "スイッチ待機中" (Waiting to switch) appears under the health bar.

This is a WIP