Pros:
-Good farmer (lots of AoE)
-Good harass/poke (Range)
-Low CD
-High damage output

Cons:
-Squishy
-Limited escape options
-Use of abilities can really make or break the experience
---overuse of abilities to burn too much mana
---lack of strategic use/placement of abilities can ruin escape and leave for dead

Ok, so assuming you're coming to this build looking for some help and already have a concept of the champion in mind, I'm just going to highlight the primary differences of this build compared to most others.

Started as a troll trial, but I really like the full Force page, 81 AP in endgame. The early scaling gives you a boon whether laning or playing mid, and likely to outscale the adversary AP. The trade off is of course, no Mag Pen and Mana Reg, I cover that base lightly in masteries but since his Q is 65% and R is 90% I feel safe with minimal Mag Pen. With a solid build of AP also, allows you build defensively early and not worry about covering AP AND survivability early game. His Q has awesome poke capability as much as farm if used wisely.

Truly, my core items are Sorc Shoes, Rylai's, and Rabadon. This is because the sorc boots again boon the Mag Pen a bit, and since Mag Pen is purely pending enemies having magic resist, unlike armor penetration, magic penetration will not do additional if penetration takes resistance to zero. So I don't make a habit of stacking too much of it. Rylai is just insane with the slow, keep chucking bombs and slowing the opponent. Then of course Rabadon for the damage output. A must.

RoA is nice too, but that is a personal preference.

Zhonya and Morello are true situational items. I like them for the AP/CD and 2nd highest AP item. Substitutions like Abyssal Scepter (Mag Res/Mag Pen), Will of the Ancients (Spellvamp), and Lichbane (Movement, Mag Res, and the proc that can augment passive) are more viable in some situations, as with any mage of course.

I've seen a few use Archangel's Staff, but I've honestly never had trouble with my mana in this build. Lots of poke/harass potential, and keeping minion waves back.

I did not discuss it, but I usually buy wards with any spare gold and place them around my lane. Usually 2-3 at a time, just to help not wander too far into a gank. If you lane with a support, a willing ally, or have a jungler, feel free to ask their ward support too if it becomes too much a burden for you to do for self. But the effort makes you that much more potent as well as protected.

Truly, I vary the spells each game based on spells allies pick and what lane I'll be playing.

I usually keep Teleport regardless, just a favored mage spell for me.

Promote helps keep the lane pushed. Trouble with it is, Ziggs is limited with his escape routes, so it's not always the best choice clearly.

Flash is nice. Another frequented mage spell for obvious reasons. Waddling yordles like it too. Stays on the Support spells too.

Surge/Ghost/Ignite: See these spells on other players, never really cared for any of them on him. Surge does Attack Speed but he's not really an AS player. Ghost... meh, can help you escape, it's not a bad spell, but he's pretty slow to begin with, you may want a head start to guarantee the effectiveness. Ignite is a close quarters spell for this ranged AP champ. Odds are if you're going in for that last hit with an Ignite, you're going to die to the other enemy champ jumping out of the bush for you. You have no stun, which works a lot better than a slow or an escape jump in this case. Sure it may be useful when fighting against a melee champ, but that really limits the spells effectiveness in how often you may get to use it. This spell is best used if laning with a melee champ, imho.

Heal/Clarity: Heal, not a bad spell, but too frequently not really life-saving. Again, with limited escape options, if you're low health, enemy knows that, and no escapes, you're an easy target even next to turret. This is especially considering solo top or mid laning. If you are laning with an ally, definitely more viable for that protection factor. Clarity, again, never had mana issues with this champ/build, so a wasted spell when more effective ones are available.

Cleanse: Never seen it used on him. I've never used it. Helpful to get out of a stun, but can't say how effective it would be on him. If you're focused... yeah, not a tank. Worth the mention I thought though, pending how much focus you receive in game.