Zermatt is one of the very best ski destinations in the world. The scenery is incomparable and there is an enormous variety of runs, both on and off piste. The mountain restaurants are excellent (there is a good food guide to its mountain restaurants).
You will get most out of your skiing if you take lessons, either in a group or (if you are very rich) privately. The instructors will show you the best skiing and it is the only safe way to go off piste.
Go to the rickety mountain restaurant near the top of the Hohtalli lift for a mid morning hot chocolate and admire the views of the glacier.
Ski the bumps on Trifti and Stockhorn.
Take the cable car to Kleine Matterhorn and walk up the stairs to the top of the lift station. It is very high and you will notice the reduced oxygen levels, but the views are worth it.
You can ski to and from Cervinia in Italy for the day (take your passport) but Cervinia's skiing is easy and dull. It does give you the chance of an authentic Italian lunch in the valley (do not use the main mountain restaurant). Make sure you set off back in plenty of time as the lift capacity in Cervinia is not great and you do not want to miss the last lift while stuck in a queue and have to take the expensive cab ride home.
Zermatt is car free but watch out for the silent electric cabs.
It is expensive even by Swiss standards
The lifts to the three main areas are in different parts of town and depending on whrere you are staying you can have a long walk to the lifts.
Interconnection between the three main areas is not great. You are probably best using the Sunegga Express lift as it is fast and Sunegga is a good starting point to explore the area.
The Gornergrat cog railway is picturesque but very slow. It is definitely worth trying to get a seat.
The piste maintenance is not up to US standards (I am British, not American, so I am unbiased on this).
Continental Europeans do not understand the concept of queuing, so be prepared to push and just accept that the tops of your skis will be scratched (I am pro US on this point).
Despite these problems, this is a truly fabulous ski destination and you should have a great time.
Here is a link to the Zermatt Tourist Office Home Page http://www.zermatt.ch/index.e.html.