We stayed in La Tzoumaz and did our day trip to Zermatt by driving to Stalden station, which has a big car park, and catching a regular train to Zermatt. It's much easier/cheaper to park. Stalden is about 15 minutes up the line from Brig. You turn off at Visp and take the road towards Zermatt. Be aware you can't drive to Zermatt so the sooner you dump the car the better. The alternative is to just drive down to Riddes and go all the way by train (Riddes-Sion-Visp-Zermatt). The Autoroute peters out just after Sion, so my guess is that this may not be that slower than driving to Brig or Stalden.
Another day-trip we've done from La Tzoumaz was further afield: We drove to Brig and parked there, then got the train to Domodossola, then changed to a scenic alpine train across the mountains to Locarno for lunch, then came back. This was a rather longer day than Zermatt.
This year, we're planning to try the Lotschberg tunnel motorail under the Eiger and then drive to the capital, Berne for the day.
However, our experience is that once you've done a longer trip, you generally want to do something more local. Options include:
Bains de Saillon in the valley near Riddes: a Large Thermal station, more family-oriented than many of the luxury thermal resorts.
Martigny: Impressive Roman Amphitheatre and if you're into it, an excellent modern art collection at the Foundation Pierre Giannada (the outdoor sculptures should interest even the kids).
St.Bernard monastery at St.Bernard Pass - but don't expect the dogs to be there (they live in the valley in the summer).
Re walks: July-August they run the gondola out of La Tzoumaz up to the Savoleyres summit, and the main gondola and chairs out of Verbier. So you have a number of walks from the summit. An obvious one is to go back down to LTZ via the attractive Chez Simon restaurant - this is 2-3 hours and all easy downhill. Pierrre Avoir has been mentioned and the view from the top is spectacular.
You can take the telecabine from Savoleyres down into Verbier just to have a look around. There's also a level walk along the contour from Croix de Coeur, below Savoleyres, around to Ruinettes.
For adults, it's viable to take this and about half way to Ruinettes, ascend to the Col des Mines above LTZ and walk down from the head of the valle back into LTZ. But don't this this in bad weather.
What I'd recommend is going up in the gondola from LTZ to Savoleyres, then
orienting yourself first, and try judging the scale of any walks. Then go down via Chez Simon to LTZ . Then you can decide what's going to be doable.