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Oh man....you may regret asking this question (at least in terms of me and info sharing!)
First, it depends on your level of comfort/ski desires and secondly, at least at the get-go of the day, it depends how you are getting up to BC from Avon.
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Getting there...depending on where you are staying, you have the option to take a bus (free) to the main BC resort area OR you have the option to take the Riverfront Gondola from the Westin up to the Lower BC chairlift and that will bring you to Bachelor's Gulch. (Note, then there's a way to ski back down to the Gondola....it's pretty nifty!). We have done both -- our preference because we stay at the Wyndham is typically to just take the gondola. BUT the bus comes often and is an easy/quick trip.
Ok...so once you are up at the mtns...there's 3 areas. Beaver Creek, Bachelor's Gulch, and ArrowHead.
BC and BG probably have the greater variety of terrain, but ArrowHead is less popular and thus mostly can be a little more relaxing.
Overall...for you,
* McCoy Park (top of the B.G. area) is a big open area of green runs. This is a nice area to just relax -- it is probably one of the flattest areas. BUT this would not be a good area to check in with the family as you can't necessarily reach each other at the chairlift.
* Red Buffalo (to of B.C. area) is another great green area with LOTS of options and a little more "steep". I love this area to just cruise and practice turns, edging, etc. This area is all green, but you'll see a bunch of different levels of capabilities because it is just such a nice area to lay back. The other good thing is it's pretty easy to get to and to meet up with others (who might be primarily on other lifts with more challenging terrain). (You can also traverse from Birds of Prey lift over there...which is a nice way to separate from the family). Note -- if your family is hanging out at Birds of Prey, and you choose Red Buffalo, it is possible to traverse back there and meet up at the lodge. The quickest way would be off of the Centennial area you can take Redtail down. It is a long steep run, but depending on how you feel, it's very open for traversing and making your way down slowly (or bombing if you so choose to get back fast!). Certainly a doable option to meet up. Otherwise, you can take cinch which is VERY relaxing, just long/slow -- which sometimes has it's merits!
* ArrowHead, Bachelor Gulch are nice for Blue Runs that are a little more challenging and yet offer some green and black for variety. Favorites for me are Gunder's, Cabin Fever, Golden Bear and Cresta.
* Strawberry Express -- I really enjoy Bitterroot as a long blue cruiser. This lift also offers black options allowing everyone can meet at the bottom. From the top you can also traverse over to Larkspur / Birds of Prey and Grouse (as well as McCoy).
* Larkspur -- I would put Larkspur as moderate/challenging blue, depending. I have a love hate relationship with the Larkspur run (depends of the snow conditions/time of year -- though as I get a little better I don't hate it AS much) and if you are avoiding moguls, stay away from Paintbrush (though it did force me to practice!). BUT the top of Larkspur also has a lovely long cruiser that you can take back down to Strawberry Express. AND from Strawberry Express you can take Sawbuck (a super fun long green cruiser with a few small "challenging" spots) all the way to Bachelor's Gulch.
*Centennial -- this is the motherlode area, though the green off of it is pretty limited. Cinch is a good way down though (or to avoid the upper steeps of the blue and then shift back to the blues lower down -- the lower part of the blues is a little less stressful).
....Um....I think I'll stop there. Feel free to reach out if I can offer other thoughts and details. I am no expert at skiing -- so, I've really spent time when we are there learning what works for my level....and it's been fun over time to see my "advance".
oh yeah....and one recommendation for restaurants in Avon, if you like vietnamese -- Pho20 is AWESOME (right next to Christie's Lodge).
oh yeah #2.....at least once you should try to be down at the main BC area to get some FRESH Chocolate chip cookies. Just look for the crowd mass following the chefs with the white hats/jackets walking around on the snow in the region mostly of the ski school over the centennial lift. They are soooo YUMMY!
Hope you have a great time! The snow was super awesome a couple weeks ago!