The Christmas period is busy at both resorts. Christmas day not as busy but the day after usually quite busy.
For a single day private group lesson you should book ASAP as all the best instructors are always busy over the busy Christmas.
As for the best beginner area of the two resorts there are a few considerations. First up if you are based in Avon then BC is just a short bus ride away. Vail a little further by bus but if early you can park your car in the Lionshead parking structure and be within 100m of the LH rental outlets, SS office and the Gondola.
The beginner area at BC is at the bottom of the mountain and is an excellent setup. To progress onto the mountain at BC is again very easy with excellent beginner terrain at the top of two chairlifts.
For Vail you have a choice of Gold Peak beginner area, again at the bottom and again an excellent setup but a little crowded at times. The Lionshead beginner area is at the top of the gondola which makes the gondola ride part of the "day out". The gondola is heated and there is a large top station that includes a choice of restaurants and a coffee house as well as activities for children.
So how to experience both resorts in two days as beginners is your main question?
My suggestion would be to take the initial lesson out of LionsHead at Vail and then progress to the nice beginner\intermediate terrain at the top of BC. Or if you have a brilliant day at LH then there are some nice beginner\intermediate trails directly accessible from the top of the Gondola. You can also ride the Gondola down at anytime till about 9:00pm at night. You can book an instructor via the LH SS and they will meet you at either resort whatever you decide or the weather decides for you. Rental skis (from VA outlets), Instructors and lift & lesson tickets are interchangeable between both resorts.
Lots of information in this post so if there is anything else I can help you with just PM me.
The past few days has seen 2ft + of snow in Summit & Eagle counties so winter is here already.