Not saying it is impossible, but students at "puppy mill" flight schools are 'normally' scheduled for 60 days to do the zero to private pilot thing, and thats all they do full time.
While I'm not shooting down your plan, you'll need to consider schedule/weather/aircraft availabilty/maintenance issues associated with your vacation plan. The CAP also has their own processes and procedures for becoming a flying memeber, usually requires several training missions as an observer before 'moving' up.