Pros


Incredibly mobile
Can kite just about any champion in the game
Great for pick making or follow ups due to her mobility combined with Charm CC
Kit has amazingly good synergy with items like Luden's, Lichbane and Rylai's
Very versatile in her build options
One of the strongest midlane gankers
"Burst" mobility makes her often overwhelming to face as a champion reliant on skillshots

Cons


Skillshots on and short range on lead to somewhat unreliable damage sometimes
Brush(and lack of vision) screw her W/R targeting over
Charms high mana cost(85) make it a punishing spell to use in lane if it misses
Naturally a very squishy champion. A caught fox is a dead fox.
Shutdown by movement impairing effects can make her an easy target in teamfights
If you have a bad game, you are next to useless as you offer less than average utility for teamfights
Extremely close range for a mage, has to use her mobility/kiting to keep herself alive when so close to the enemy

For Ahri I always use movement speed quints as I like to focus on the moving part of her kit, whether its for roaming or kiting, running fast is great! Combining it with many other MS enhancers can make Ahri ridiculously fast not even including the mobility she already has in her kit. That being said there is absolutely nothing wrong with choosing to go for AP quints if you want the damage. It's a preference and playstyle choice here.



For mages there is only 2 choices for marks, Magic penetration and Hybrid Penetration. Which depends on the player, the difference in magic penetration between the 2 sets is only 2.3 so basically depending on the player/champion IF you autoattack your laner whether it be harass or in duels hybrid penetration will work out better for you. Flat penetration is for those champions that don't ever auto (Karthus / Cassiopeia) but Ahri is one of the mages that fully utilizes autoattacks at all times so she will benefit more from Hybrid pen.


The purpose of these flat health seals is just to keep your level 1-6 as smooth as possible, an extra 72 HP is pretty significant especially early on to keep you decently durable to survive against lane bullies or champions with strong harass.
If you are against a laner who deals physical damage, you can swap 5 of these(or all of them) for flat armour seals as it can be somewhat suicidal to face most of them with no bonus armour. AD laners tend to be quite scary with their all in's and autoattacks that hit as hard as spells.



Most of the time I'll run CDR/level Glyphs on Ahri since I don't often get any CDR from items early in favour of Needlessly Large Rod items. Running CDR here is IMO the best option since you lose the least for picking up the CDR in your runes.
If you are against something scary like Annie or LeBlanc I'd suggest you run Flat MR glyphs instead, 30 MR vs those people is a deathwish.

Offense

Since you are a soft mage whose primary focus is on dealing damage, offense is the tree to dedicate your points into since it obviously will offer the most benefit towards your damage dealt which is the most important for you to try and improve.
A common argument for this mastery is that it is bad on ranged, because if you pick it you are giving the enemy the mastery as well. Now that is sort of true but not entirely since in a teamfight you can hit as many as 5 people but they can only hit 1 of you. Not only that but when your primary role is dealing damage, that takes priority over all else including your own survival. Finally, 1.5% on a soft squishy is nothing, everything hurts a lot it is just whether or not it kills you that matters and 1.5% extra damage taken is really not going to be noticeable.
This is an absolute must since you shouldn't ever have 40% CDR from items alone, so any free CDR that can count towards that cap is worth having.
Any ability power masteries are going to be worth taking on a AP scaling mage.
5% extra damage against someone under 50% health is HUGE! This mastery is awesome since 5% is a pretty big modifier and 50% health is a massively generous health threshold for an execute.
This mastery is pretty much a must have for any champion in the game that will buy a , 5% AP for 3 points is really amazing, looking at it this way if you have 400 AP from items alone you get 20 AP for free from this mastery. Thats a free 435 gold! making it more valuable than the Mental Force mastery and it only scales further.
This mastery is pretty awesome for only 1 point. It might help you in lane by saving you from an Ignite if you just killed an enemy champion but its really value lies in teamfights, the potential to regenerate up to 25% of your health/mana can easily make the difference between surviving it or dying during it.
Penetration is a very powerful and difficult to obtain stat since it is so rare. This mastery is a must on any damage dealer in the game.
This is a quite a good 1 point wonder. Any damage helps and this especially helps with last hitting as the game goes on.
If you look at the executioner mastery, which is a good mastery, it works out to be a flat 2.5% increase in damage, generally. 2.5% for 3 points and this is 3% for 1 point, simple maths says it's a good choice especially since you already went 20 points into the tree.

Utility

Utility is the tree I dedicate my last 9 points too. Reason being is the 9 points into utility just offers more use than 9 into defence. The added benefits of lower summoner cooldowns, longer blues and movespeed just outweigh a small amount of extra durability.


Since you will be using trinkets all the time you might as well make it have a bit more range, this is especially useful if you use the ward trinket to scout a bush that you think someone might be in, that extra 15% might be the difference between you getting garen'd or not.
Movement speed is awesome, this is basically a free movespeed quint and basically moves your champion up a tier in base movespeed.
Having Flash and Ignite up as often as possible is important for playmaking and in general having more uptime on them.
A point to make potions better. Only an extra 15 hp on a health pot but you need the points to get to the buff duration mastery and this is probably the best one for 1 point.
I prefer to run Culinary Master on Ahri since I tend to get involved in trades a lot, and I buy a lot of potions so I will get the value out of the extra 20 health/10 mana per potion.