well, I ski a lot in Switzerland since I basically live there. Here is some advice:
Do not go to Engelberg unless you are really rich... It's a rip off. True, they have some nice challenging backcountry runs but they are among the most expensive ski resorts in Switzerland. Plus, Engelberg can get very very crowded - especially on the weekends since Luzern, Zürich, Basel and Bern are not very far away. I usually avoid Engelberg unless I'm out touring on skins, etc.
St. Moritz is very chique. They are having a very bad winter though (most of Europe in general, but especially Graubünden, Wallis and the southern Alps). You can ski down to the village in St. Moritz at the present - but only on artifical snow. The terrain is great around St. Moritz if you can afford going heli skiing or touring. Otherwise I think it isn't any better or worse than any other resort - just different marketing - so St. Moritz is also quite expensive (compared to the prices in the US it's probably still quite reasonable).
Meiringen Hasliberg is a small ski resort with lots of T-bars and a few chairs. It's more suitable for families with little children. For easy skiing it's OK, but if you're looking for challenging terrain don't go for it.
Grindelwald is definately great for the scenery. The ski area on the Scheidegg is OK, but lots of tourists (Japanese, Americans, etc...). As already recommended the best skiing in that area is from Mürren. You ski on the Schilthorn (that's by the way the scene where they shot a James Bond movie). The Schilthorn is great. It only has a few lifts but a big big vertical. So you get lots of skiing for the amount of using the cablecar. Make sure though before you go to check whether they are open down into the valley.
Villars and Chateau d'Oeux I haven't skied yet. Les Diablerets is on the other side of Chateau d'Oex. That's quite OK. So I think Chateau d'Oeux should work out well, too. I haven't skied it though.
Absolute insider tips are the ski areas in central Switzerland. They usually get the most snow. Andermatt of course is killer. I ski there a lot. It has 2 cable cars up the Gemsstock. You leave Andermatt at 1400m (4200ft) and exit on top of the Gemsstock at 3000m (9000ft). That gives you a vertical of 1600m (4800ft). Andermatt is very steep and allows great backcounty skiing which is easily accessible from the cable car. They also have a few lifts but usually we only use the gondola. Andermatt is definately worthwile going.
Then there is also Hoch Ybrig. Ybrig is quite small and doesn't have the really challenging terrain. It's more like Meiringen but on fresh powder days it's fun since you have quite a few possibilities for "small" backcounty skiing.
Then you should still consider the Wallis. Especially Verbier, Champéry and Saas Fee. Verbier is a little more on the expensive side but it has absolute killer terrain and it's HUGE. You can do anything there from easy cruising runs up to extreme skiing (Mont Gelé). Unfortunately they are having very little snow at the moment. Champéry (Portes du Soleil) is also very cool if there is enough snow. You can easily access nice couloirs and steep backcountry. Besides it has the largest groomed slope network in Europe (or the world?) with over 650km. This doesn't mean though that you won't find any backcountry. But they too are having very little snow. Saas Fee is very big. It goes from 1800m (5400ft) up to 3600m (11000ft). This gives you a vertical of 5600ft. It's not as good though for tough backcounty skiing. There are some options but it is a little limited (due to the huge glacier with all the crevasses).
Hope I could help you. If you have any questions - email me!
And yeah, St. Anton is always a good bet. They are having pretty good snow right now compared with other ski resorts!