Ahri is basically a mage with mobility. She has an average amount of AP damage output, high mobility, a hard CC and a sustain, and is fragile. She's solid all game long, but best midgame. If she get's ahead early she can become a monster midgame and carry a team. She's notable for being a pick that works in almost all situations.
Ahri's biggest weaknesses are the short range on some her spells and that most of her mobility is tied up in her ultimate, which has a cooldown. She is also skillshot based, but accuracy in skillshots is less important than having good judgement on when its safe to get close to an enemy for a kill. That's really only something experience will teach you.
At laning phase, she primarily wants to throw her Q and run, occasionally targeting the creeps when her passive is up. Ahri has reasonable harass, mobility, and sustain, and depending on matchup will leverage these to win her lane. I don't find her big on getting kills in a lane myself, not unless the enemy is foolish and leaves their tower with less than half their hp, so usually focus on cs, not getting ganked, and pressuring your opponent out of cs. I don't tend to gank before 6 either.
Midgame she can roam and apply gank pressure. She can also push fairly safely so long as her ultimate is up. Ahri's damage is particularly high at around this point so if you're doing fine you can quite aggressive. Do not be vulnerable when your ultimate is on cooldown.
Lategame she plays Bait and APC. In a teamfight she prefers to be a backline APC that dodges divers with her mobility, or cleans up a teamfight with it. She can also bait very well, as she is a preferred target but also slippery. You can win games against enemies who chase foxes.
For getting the most out of her skills, you can use Q as your typical workhorse, but save Charm until near certain it will hit. It has a much longer cast time and cooldown, and people will try to bait it out of you if they can. You don't need to hit an enemy with it to hit them with Q and W, even if it feels nice to combo. Once you activate her R, know that you don't need to hit the button again immediately. You usually don't need to burn all 3 dashes at once, unless you're doing so to do as much damage as quickly as possible. Finally remember they ARE dashes - flash is faster. That can save your life.
For building her, she can make most mage items work for her. My own standard build is:
21/9/0 (since the meditation change, I gave up on 21/0/9 and built double Doran's instead)
Armor seals, MR glyphs, Mpen marks, AP Quints
Start:
2x
First back at 1350g:
1600g:
, if pressured before then head for
,
if you don't have them
or 
Note that your w counts as single target for Rylai's. The most common adaptions are to swap
for
, against certain AP damage dealers and point and click stunners such as Annie.
and
together can be used against tank heavy teams with guys like Cho'gath and Mundo. Against particularly squishy damage focused enemies I sometimes get GA before Deathcap. Zhonya's can be swapped in against certain enemies as well, but I tend to find GA does the job of being the defensive item better.