I personally main Ziggs as my AP mid and I've yet to go wrong with him. He's fairly hard to master because his combo requires precision to pop off properly, worlds away from a combo like Leblanc's or Ryze's.
Ziggs main attraction is his insane range with his bouncing bombs. While there is a range indicator when you hover over them, this is deceiving as it shows the maximum range for the initial throw - the bomb continues to bounce three times in the direction you threw it after this throw, making the range quite long. Because of this Ziggs' bombs make the perfect harassment tool during laning phase as the only way for an enemy to escape them will be to dodge or move out of their long range, denying them CS and possibly XP if you zone properly.
During laning phase I'll usually start with the standard boots + 3 pots. I'll aim for lots of last-hits, using my bombs for harassment so my passive (bonus damage on autoattack) procs more often (cooldown is 12 seconds but it is reduced by 4 seconds for every spell you cast). I'll usually max bombs by the end of laning phase, with my minefield a close second for the CC (the C4 charge is for utility and is maxed last). I'll pick up a few kills from jungle ganks during laning phase and if I'm doing well by its end I'll have picked up a Rod of Ages (for some tankiness and more mana for bomb spam), a Hextech Revolver for sustain (build into Will of the Ancients if you have the gold), and boots (usually go with Sorc Boots for magic pen).
During mid and late-game Ziggs main goal is to win lanes by dropping his combo whenever a lane is being pushed in order to clear out anyone camping at towers. First, drop your minefield BEHIND the person you're targeting (I cannot stress this enough - dropping it on top of him allows him to flash out of it or to see it coming and move back; dropping it behind him catches him off guard as he attempts to escape and runs into all the mines, being slowed). As soon as the animation shoots down, drop a bomb right on or close to that person (you can take advantage of the bounces to be able to move a bit away in case you're afraid they're baiting you), and begin chasing. Between spells, make sure to hit autoattacks as they will do more damage thanks to your passive. Continue dropping bombs and if he attempts to flash, drop your C4 charge and blow it up to catch up with him (remember: Where you stand determines in what direction you'll jump!) and pick up the kill.
Your ult is quite special and only for use under certain circumstances. When dueling characters, if you know that you cannot escape and you must drop your ult to survive, make sure you do so a few seconds AHEAD of when you'll be low on HP. The drop is delayed even if you target the ult right on top of yourself, so if you shoot it off when you're low you'll die before it lands. Otherwise, never use your ult for single character kills as its much better for targeting a cluster of enemies (useful for following up a hard CC such as Amumu's which forces all the champions to stand still in their cluster, leaving them at your mercy). Only exceptions to this rule are for saving teammates and killing the carries (both AP and AD). As always, the time and place when it's proper and not proper to use your ult all depend on the situation. What's key is to ask yourself, "Will this ult do maximum damage and accomplish something that my other abilities cannot due to range or damage restrictions? If so, will this accomplishment help my TEAM or only myself?" If the answer to the first question is no, or if the answer to the second is "myself", don't use your ult.
Now, as for late-game items, you'll be looking at investing in Rabadon's Deathcap (the standard AP item for most mids), Crystal Scepter (for the slow and for more tankiness), and possibly a Void Staff if the enemy is stacking Magic Resist. Items worth looking at as well include Lich Bane (to stack with your passive) and perhaps Abyssal Scepter (for tankiness). The end-game build really is up to you but it should emphasize having lots of AP (of course) and, if possible, some magic penetration and tankiness (Ziggs has to be rather close to enemies to pop his combo, so it's not unheard of to be bursted while doing so and a bit of health doesn't hurt).