Well, you're in for a treat! Zermatt was one of the most beautiful places I've been. Depending on how "close" you are to the train station, it still may be a bit of a walk to the closest lift. It's actually a funicular and then you upload to a gondola (Alpen-Metro -> Sunnegga). Not terrible, but could be an interesting walk back at the end of the day with 4 tired children
Unless of course you upload/download via the Gornergrat which leaves very close to the train station.
Good choice on not bringing skis. The transfer in Visp can be a nightmare with multiple bags, especially ski bags. Pretty much a stampede from the main train to the cog train you need to board and gear storage is limited (relatively speaking). We only bring our boots when we go to Europe, less hassle. Although, you're going during a non-peak time so hopefully it's not too bad.
Hard to talk about the skiing in Zermatt without writing a novel, so just grab a map and go. There is plenty of terrain for all abilities and perfectly groomed. I would most definitely recommend the add-on ski pass to ski into Italy, it's beautiful! Keep an eye on lift statuses because the connection is often closed due to wind/bad weather. Not that the add-on is expensive, but perhaps it makes sense to buy your passes daily so you can determine when you want to go to Italy based on schedule and weather.
I'm not sure if this is feasible since you'll be traveling with children, but try to hit Cervo for Apre. It's a beautiful and lively spot at the literal end of trail when you come back down through Sunnegga. This assumes there is enough snow below the tree line to get there. They just got a refresh recently and they have decent snow making on their lower trails so I'm sure you'll be fine in early April, provided no major warm-ups.
On Mountain 'must-stops' :
- Stafelalp on the Furi side
- Chez Vrony on the Sunnegga side (off of the Findlebahn)
- Fluhalp on the Sunnegga side
- Findlerhof on the Sunnega side (off of the Findlebahn)
- Iglu Dorf (off of the Riffelberg) - neat spot to check out (especially with the kids) the huge igloos and they have mulled wine
- Adler Hitta
Soups and Charcuterie were the main go-to's when we stopped at those locations. Don't sleep on the Rosti at Alter Hitta, however.
My wife and I don't really like the 'pinkies out' style places when we go skiing, and there's certainly a ton of them in Zermatt. So we tended to find the lower-key spots to eat for dinner. We frequented Grizzly's quite a bit (near the train station) - a low key bar/bistro. Another one we hit was Du Pont, in the middle of Zermatt. Your classic fondu place )with other options) and apparently the oldest restaurant in Zermatt, but still very low key and unassuming. If you're looking for Italian, Refuge du Théodule!
Overall, the food in Zermatt is
fine, but don't expect to have your socks blown off. You're there for the magnificent skiing and scenery, not the food
When compared to other resorts in Italy or France, it certainly doesn't rate as high, but that's expected. With that said, make sure to stop at a place or 2 on the Italian side to indulge
For gear rental, we went to FLEXRENT which is "near" the train station, very very close to Sunnegga. You can also drop your skis off their at the end of the day as well to hopefully make your walk back with 4 tired children easier lol We stayed at Hotel Schonegg just up the hill from FLEXRENT so didn't need to drop skis off but always the option.
Hope this helps a bit!