Let me start off by saying that I am by no means a veteran player. I am currently lvl 24 with around 100 wins, but I have fallen completely for Ziggs. I would estimate that I have played around 150-200 games with Ziggs, and I feel I have learned to master this champion.

The way I play Ziggs makes him a mediocre 1v1 champion but an invaluable asset in teamfights, landing many killing blows and assists.

As AP mid I always start with boots and potions. This will not maximize your damage output early in the game, but it will help you stay in your lane for as long as possible without having to recall to regain health or mana. If you play against other champions who have alot of skill shots then you will see that the extra movement speed you get from the boots are invaluable.

The build order I have set up is not one you should follow 100%. I sometimes build Rabadon's Deathcap after I get my Sorcerer's Shoes if I need the extra movement speed early on.
Getting Needlessly Large Rod or Blasting Wand first depends how much you have farmed when it is time to recall. If you have HP and Mana and potions left, do not recall just to buy the Blasting Wand.

The other items I choose for Ziggs are there to maximize AP. This is a "proper" glass cannon build. The only defensive item you need is Zhonya's Hourglass which gives you armor and the ability to make yourself invounerable for a brief moment. This ability should be saved for when you are beeing focused in a team fight, or if you are beeing dived and you think the tower damage can help you kill your foe.

I think all AP mid Ziggs guides have the same skill sequence, so I will not spend alot of time explaining this.

Q (Bouncing Bomb) is your main damage dealer. Max this ASAP!
W (Snatchel Charge) is your utility spell. Get this at lvl 4, max it last.
E (Hexplosive Minefield) is a great damage, utility, slowing, area denial spell. Get this at lvl 2, and max it second.
R (Mega Inferno Bomb) is your ultimate. Always get this when you can.

Pros
+ Great range
+ All abilities deal AOE damage
+ Great escape/gap closer (Snatchel Charge)
+ High damage burst
+ Very strong late game
+ Can turn a team fight with Q -> E -> R
+ AOE damage slow
+ Awesome skins

Cons
- Squishy
- Weak early game
- Mana thirsty early game
- Needs blue
- ALL abilities are skillshots