This guide is intended for those of you who want to try out Ziggs and want a quick reference guide for the basics and is for summoners rift or twisted treeline. This is not a guide for someone who wants their hand held. Ziggs has a bit of a learning curve but is very good solo and group fighter. He can slow, knockback, deal very good aoe damage and his ult can even steal baron.

Marks: Magic Pen
Seals: Mana per 5 sec (Ziggs needs to be able to harass and spam without taking dorans ring)
Glyphs: AP per lvl (scales well and as good as flat ap by lvl 6)
or CD redux (i prefer AP per lvl since it scales well and is great late game esp with rabadons)
Quints: AP

Standard Ranged AP burst set up, no surprizes here.

Obviously the later game items vary according to the circumstances of the game but the core items i always build are:
2 health pots + Mana crystal: 200 mana and the ability to regain health is going to give you an advantage over the players who take a dorans ring first. Also it builds into a catalyst the protector after you get your boots of speed.
Boots of speed: obv next choice
Catalyst the Protector: At this point the advantage you have over the people who took dorans ring first is huge. Not only do you have a boost in health and mana, you get a huge health and mana regen passive that allows you to push and harass more freely. And you dont have to rush the ap item component of the Rod of ages before this because you have AP per level glyphs.
Rod of Ages: get it asap
Sorc SHoes: magic pen obv.

The next choices depend on the game. If you are doing well, and there are no problems then I go:
rabadons -> rylai's -> deathfire grasp -> void staff
If they are tank heavy then i rush a deathfire grasp or at LEAST get Kage's Lucky Pick to help move the build along more quickly and push rylai's to the end of the list unless the tank is singed.
The only other items really to consider are all game state oriented and depend on the circumstances. For Example, if you are mid and you are doing well mid, have your rod and boots, but your mid opponent is out pushing you, then you might want to consider getting fiendish codex. It has everything you need to push your lane and also can build into a deathfire or a Nashor's Tooth. (Nashor's would not be something you would get often, but something that is very effective when you are constantly pushed to your own turret.)

Honestly, there is no other skill sequence needed imo. Some people might wants to go q -> e -> w but I really dont find it necessary to take w any sooner than level 4 because the threat of gank prior to level 4 is low and you obviously have flash as your main escape and it is most likely off cooldown so there's no reason to gimp your main attack spell and delay yourself a level just to grab a utility spell 1 level early. I cant think of anything to make the case for taking e first, however I have seen at least one person do it and if anyone wants to make their case on that I would be glad to hear you out.

Flash is the #1 choice.
Ignite is #2 and most often used.
Others to consider over ignite depend on your team composition and/or playstyle.
Clarity is good if you dont have mana regen seals or if you are laning with a support char that didnt already take clarity.
Exhaust is good if you are laning with an ad carry or if your tank is forced to take say teleport and ghost and no other team members have exhaust or even if you dont have a tank.
Ghost can be taken over flash if you want, but this is one of the only champs that I always take Flash over ghost just due to the fact that he has built in escape options and the flash gank or flash in the jungle is more appealing than the sustained speed imo.

Playing Ziggs is all about anticipating your opponents moves or forcing their moves according to your mines and charges. Make sure to remember that your bomb bounces, so you can make use of that extra range while people are running away or hiding under turrets. Save your charges! You need your charges to escape or knockback enemy champs to save an ally. You only want to use them when farming if you are pushing and have eyes on all the other champs and they are no where near you, or if you are in a group trying to finish or prevent an enemy escaping. Mines can be used to prevent escape or prevent pursuit or even to throw in the bushes by certain jungle exits/entrances by turrets to be like a blind ward and trap. Your ult has a decent range, it's not instant and requires practice to get the timing down. Try to remember it's like being a quarterback and your target is a reciever. You want to throw your ult to where the champ is going to be while considering the delay according to how far you are away. You can also use it to blast champs/minions while trying to save a turret that is unattended. My personal fav is dropping it in group fights after an Amumu stun or galio taunt.

There you have it. A basic guide for the clinically impatient :)