dz,
Chamonix is definitely the spot for "serious alpine hiking". Yes, Hôtel l'Héliopic is very close to the Aguille du Midi cable car station, but you can stay anywhere as the town is very walkable.
I'd suggest having a look at a couple of guidebooks to determine the "must see / must do" items that appeal most to you. If you're more adventurous and your budget will allow, you could always do some tandem paragliding while there.
IMO, a trip to the Aguille du Midi definitely belongs on the list and at the top you can stand on the glass "Over the Void" platform and look at the view many hundreds of feet down to the rocks! However, note that the top level is at about 12K feet, so if you have any "altitude issues", that could be a problem.
You could also take the Panoramic Mont Blanc across to Helbronner (Italy) over the glaciers, and that lift system is one of the longest in the world without support towers. From Helbronnner you can either just stop there for awhile and have lunch, or go down to Courmayeur for a brief look. There are lots of options in that area, so your only limits will be time and money.
Another activity is Montenvers and the Mer-de-Glace ice caves.
Regarding a "day trip" to Zermatt, as a previous reply indicated that's not really practical due to the travel times involved. The trip is about four hours each way with a minimum of two changes. Also as mentioned, you could find the Matterhorn shrouded in clouds when you arrive, which would make the entire trip useless.