I don't know if this needs making, but i sure wish i had something like this back when i started as i struggled a lot to figure out how initiative works, who gets to go first in combat etc.

Move orders
Before we talk about initiative we have to know what the basics of the combat system are. In your ninja tab, you can see ''Move X'' above each of the characters in your formation (X being 1,2,3 or 4).

They are determined like so: the characters in the front row (the right one) will have prior moves and if you have 2 or 3 characters in the front row, the topmost one will have move 1 and everyone below them move 2 or 3. Then the same goes for middle row and the back row (left one). This doesn't mean you have to place your character in the top-right corner of the formation for him to be the fastest, but rather move one is given to the character who is the right most and then top most in your formation.
In PvP combat the attack sequence will go like this:
- Your or opponent's character who has move 1 will attack first, decided by the amount of initative they have
- The slower move 1 character will attack
- Your or opponent's character who has move 2 will attack, initaiative based
- The slower move 2 character will attack
...
- Clones attack
So, basically character with move 2 can never use standard attack before character with move 1 no matter the initiative (if they aren't debuffed).
What is initiative?
http://i.imgur.com/c2vgJ6b.gif
Initiative measures just how fast your characters will act in combat. In other words characters with greater initiative will get to attack/do their ability before the opponent's character with the same move number. You can check the initiative of the character in Character -> Details:

That sounds great! How can i get more?
You get more initiative on your character via refining. For refining you need refine runes which you can get from various events, Group and Ranked Shop, Sage World Battlefield (or as i like to call it Salt World) or Great Ninja War packs. You have 3 tiers of runes:
- Low-Level - refines equipment level 1 to 4
- Medium-Level - refines equpment level 5 to 7
- Advanced-Level - efines equipment level 8 to 10
1 rune gives you 1 wish point and the more wish points on a piece of gear you have the better are the chances that the next rune will upgrade the level of that piece of gear. Each of the 5 pieces of gear can be refined up to level 10 and each level yields more initiative.
Mysteries
If you ever wondered who gets to cast their mystery first (like i did for so long) let me explain:
There are 2 types of mysteries - prompt and non-prompt. Prompt mysteries take priority over standard attacks adn will activate before any standards attacks of the round while non-prompts just replace standard attacks.
The promps are ordered just like regular attacks so characters with move 1 will have the advantage with the one with higher initiative will go first, then characters with move 2,3 and 4 and then the standard attacks and non-prompts happen.
Note: if you activate a non-prompt after that character attacks he will use it next round instead of his standard but the crhakra deduction will not happen in that round.
What this all means?
So, everything so far has been game rules and theory, this is how you apply it.
Formation and Initiative
In most of your teams you will have 1 character with a combo starter on round 2, whether it be your main or someone else and in most cases you will want that character to have move 1 in your formation. This may be risky sometimes as the high damage characters usually tend to be on the squishy side however you can cover them with clones or protective characters (who have the passive of being the only attack targets in the front row). Good exaples of these are Neji, GNW Neji, Hinata, GNW Kankuro's puppet (not the best idea as you want it alive rather than facetanking).
In addition to this you want the highest possible initiative on your combo starter in order to be faster than the enemy character with the same move. I recommend putting all your runes toward maxing the gear level of that character, especially during the refine events.
In combat
Since most characters will want to start their mystery on round 2 you have round 1 to compare - who has the highest Initiative, you or the enemy. If you attack before the enemy, it means your move 1 character has higher initiative and it's safe to activate the mystery. However, if the enemy attacks first that means they will activate theirs first. In this case you might want to hold your mystery. If the enemy has any source of acupuncture, immobile, chaos or interrupt in their chases your mystery will not happen if you activate it, you will lose chakra and it will be put on a cooldown. Such sources may be - Hinata, Shikamaru, Ino, Ningendo, GNW Neji, Water Main, Hashirama, Ao (all i could think of so far, do tell if you know of more). Not only this but the AoE paralyze of Lightning main shuts down your combo entirely even if you manage to use your mystery. In these cases it's better to hold your mystery for after the debuffs apply and either wait for them to cleanse if you have a debuff remover or use a different combo starter. Of course this also mean if you have the higher initiative and any of these chracters you can try to interrupt the mystery of the enemy which will give you a big lead in the
Neji, Iruka, Shurado and the bunch
I've seen many people put these guys in the back and then try to interrupt the mystery of front line enemies. Hopefully after this text you will understand that can't happen so chose your targets wisely or put them in the front.