It's recommended to get one of your spells to rank 5 as quickly as possible to benefit from its Transcendent bonus.
To get out 4 bonus spheres (the maximum possible) requires more than 30% cooldown reduction and the following combo: Q, Q, W (targeting the first sphere), Q, Q, followed immediately by Unleashed Power. However, the delay on her ultimate and the increased micro management requires a higher level of skill on an already difficult champion. For most players, it may be more efficient to try using Q and E, Q again and W on the first dark sphere, and then use Unleashed Power - this sacrifices one or two dark spheres on her ultimate for a quicker, more reliable combo.
Dark Sphere has no channel upon cast, which means you can cast it without interrupting your movement. Take this into consideration when playing Syndra, since it allows you to kite and chase enemy champions easily.
To maximize your chances of hitting the enemy champion with Dark Spheres, use it right when they last-hit.
After you hit an enemy champion with Force of Will, follow up with Dark Sphere; they'll be slowed and have a hard time avoiding it. If you are confident enough in your own aiming, casting Dark Sphere right after Force of Will will cause them to hit at the same time; this can save a few seconds if you are considering a combo for Unleashed Power.
When being chased in the jungle, use Force of Will on a nearby neutral monster to slow the enemy champion.
Try to stay back in teamfights since Syndra is pretty fragile. If an enemy comes too close, punish them with Unleashed Power. In case that enemy doesn't die and continues fighting you, use Scatter the Weak right after your ultimate to stun him/her. It may turn the situation around and let you flee or kill that enemy.
Syndra's damage output is reliant on a high CDR build - it will provide stronger harrass with Q and W combos, and being able to maintain more dark spheres on the battlefield will potentially increase the damage on Unleashed Power.
She is extremely mana hungry. Athene's Unholy Grail provides every stat she needs; AP, CDR, and mana regen.
The passive on Rylai's Crystal Scepter synergizes with all of her spells to varying degrees; however the bonus mana on Rod of Ages will help with Syndra's high mana costs and may be a preferable option depending on your runes and masteries.
Building a Deathfire Grasp will help provide Syndra with more AP, as well as its active effect will help with her damage out put and improve her burst.
It has been debated that Manamune is a better purchase on Syndra due to the Muramana active, which triggers per sphere during Unleashed Power. With only Muramana for mana, the bonus damage per sphere at full mana is between 93-129 (level 6-18). This is the single most damage attainable from only a single item, and is the equivalent of having 465-645 bonus ability power. This is 279-387 bonus damage with only the minimum 3 spheres.
If Syndra keeps creating Dark Spheres in a situation were she would be better off doing something else (like re-positioning or knocking back enemies), then she is likely preparing to use her ultimate.
But beware when, in teamfight, she has 3-4 Dark Sphere on the terrain, because she can then deal massive damage with Unleashed Power (up to 1260+140% AP), followed by Scatter the Weak to stun the whole team if you're not spread enough.
Syndra is a bursty mage, try to focus her down first if you have problem with her damage.
In late game she can be very powerful with her spammable nuke. Try to deny her farming early game to cut down on her damage potential later on.
The Scatter the Weak and Dark Sphere stun combo can be very hard to land if enemy is too close to her, meaning bursty melee assassin can kill her with ease.
If using this method, beware her ultimate Unleashed Power since it has pretty short range, meaning she might use it on you, being the closest person. Try this plan when her ultimate goes on cooldown. (Warning: at rank 3 her ultimate's range is increased by 75)