Advice needed: Skiing in Europe for the first time....

I have skied in the Innsbruck area, however not in Switzerland. The Alps has an incredible number of ski resorts, however they're different for those in the U.S. For example, one ski mountain in the Dolomites (Italy) has more ski runs than in the whole State of Colorado. You can ride a lift up and ski down to one town. Then ride a lift up and ski to another town. You can do this all day and catch a bus back to where you started the day.
I would say the European slopes are numerous, but many are steeper, not as wide and more difficult than slopes in N.A. European slopes can often be icier than slopes at some North America mountains famous for powder skiing. Also, our resorts have more snow making capabilities than resorts in Europe. Of course every year is different in the mountains for snow.
As far as your abilities to ski in Europe. You can always find some slopes on a mountain to ski, however if you're just a green slope skier in the U.S., you may not be able to ski the majority of any ski resort in Europe. There again, every ski mountain is different. Most strong blue slope skiers will do fine in Europe, however. I am amazed at the double black diamond skiers in European resorts--many more great skiers than in the U.S.
When we're wanting our travel fix of the Alps, we usually fly into Munich and go down to the Innsbruck area. It's easier to travel through, and much cheaper than Switzerland. And the skiing and mountain scenery from the top of the mountains is absolutely incredible. In the U.S., most of our skiing has been in South Lake Tahoe and sometimes Aspen. I have given up skiing, however because of an accident breaking both legs 10 years ago.
Good luck on your trip.

sarah, I've not skied in Europe (had plans to do so, but knees said no). So I can't answer your questions, but in case you dont get good input here, talk to someone at Ski.com They are a travel agency that packages ski trips everywhere including Europe. I found they had good information. My impression from my previous research was that you can generally expect fewer groomed runs, little to no snow making, and a less protective nature. Hopefully someone here has better advice.

You can take my suggestions with a gain of salt in that we are cross country skiers. We enjoyed our 10 days in the Davos-Klosters area. The joy of skiing in Switzerland is both the skiing itself, skiing infrastructure, scenery and the after skiing ambiance of the ski villages. During our stay in Klosters we took a rest day for a Bernina Express sight seeing rail excersion to Italy for lunch and back to Klosters in time for dinner. Our hotel visitors pass allowed us free train travel between Klosters and Davos.

Some web sources/articles:

https://www.j2ski.com/ski_resorts/Switzerland/

https://www.planetware.com/switzerland/top-rated-ski-resorts-in-switzerland-ch-1-2.htm

A Swiss friend from Zermatt talks about very long all day ski runs from Zermatt into Italy then taking the train back to Zermatt. I don't know the details of how, but you will need both stamina and a lot of stops to enjoy the view. Zermatt is car free. Note that the J2ski website reports the total length of piste runs. For example Davos-Kloster = 320 km + 307 km = 627 km of piste runs. Zermatt = 394 km of runs. You will not get bored. Good thing you are going with gravity and not against.

Yes to bringing you boots that work for you. You need comfortable boots that fit for the long runs. Legacy carriers like United and Swiss Air include one free checked bag per person on standard economy fares. One duffle bag between the two of you should carry two sets of boots, helment padded with ski clothing.

We spent our first three nights in Luzern (flying in to Zermatt and taking the one hour train to Luzern). I don't think that Luzern is near ski slopes for day trips from Luzern, but Luzern is a good place to rest from the onvernight flight and adjust to time zone shift.

I would add one more caveat, especially because you are a green slope skier.

The ratings in the US are relative to that mountain. They are not standardized. That means a blue run on one mountain would be labeled a green run on another. For example, a blue at Waterville would be a green at Cannon. Many skiers have gone on a green run on a new mountain and found it much harder than the ones at home. That’s because the entire mountain is harder than the mountain at home.

It’s important to look hard at the ski maps.

The area of the 4 Vallees in Switzerland has a wide variety of runs at all levels of difficulty. The main difference between skiing in the US and skiing in Switzerland, or much of the alps, is that it's a mix of skiing on glaciers and mountainsides. If you are up on glaciers you must pay attention to markings and trails for safety sake. Mountainside runs are also well-marked and easy to follow.
The best thing about this ski area is how incredibly vast it is...it links 6 ski resorts by a system of gondolas that all work on the same lift ticket. You can ski for a month and never take the same route twice! It's also fun to just ride around from peak to peak...the highest bar in the world is at the top of the lift at Mont Fort. Not a place I would try to ski down from but the views are spectacular. The whole area is pretty incredible.

On the website https://www.zermatt.ch/Bahnen-Pisten/Panoramakarten-Facts/Pistenplan-Karte-Winterpanorama you can download a slope map of Zermatt. However, there are not a lot of blue slopes. If you want blue slopes, you might be better off in smaller (and cheaper) ski areas.
Just to throw out an idea: you could base yourselves in Lucerne and then decide on the day whether you want to go skiing, depending on the weather. You can reach Sörenberg/Rothorn, Melchsee Frutt, Hoch Ybrig, Stoos, or the more difficult ski area of Titlis as a day trip. You can also just buy a half day pass in the ski areas after 12 o'clock.

Thank you guys and gals for your advice! We have already booked a room in Zermatt for 4 nights, so we are definitely going to be there and then I think the next three nights in Lucerne. We may not ski in Lucerne, just take it easy and have a resting part of our vacation.

I understand what one poster said about the color ratings in the USA being all relevant to the mountain. My husband can ski black diamond in New Hampshire but not Colorado for instance. He also swears that I am a better skier than I think that I am, I just need to be more confident.

I have studied the map of Zermatt and Cervinia more, it does look like I am just going to need to get confident and go on some red slopes.

Thanks for the advice, I am going to look at the links that were provided and will be interested to read any more helpful advice. Have a great day!