The age to start training...
Training a puppy in a class situation is best to start around 9 to 12 weeks of age.
Most schools prefer your pup to be given a series of 2 vaccinations before starting in a class.
This is for protection of your puppy and the protection of the other puppies in the class.
Needed beforehand are a Bordatella Vaccination (type of kennel cough) and a negative stool sample.
The stuff you teach them at home...
Home training should begin the day you bring your puppy home.
That doesn't mean training needs to be intensive right off the bat.
At this stage in training, you should work with your dog in short spans of time (5 to 10 minute sessions).
From there just easily bump up the time.
Dogs and puppies are instantly learning from birth...
At first their world is small.
What they learn comes from their mom, the other puppies in the litter, and their environment.
By the time they are ready to be sent on their way to new loving homes, they have learned a cast amount of things.
All of this from their mom, litter-mates, and their environment.
Hopefully, if the pup come from a mix-breeder, the pup has had a good foundation set for future learning.
That is why it is good to buy a puppy from a knowledgeable and reputable breeder.
Leaning comprises many factors:

1. Socialization - Positive exposure to different people, environments, situations, sounds, etc...
   
2. Innate personality - All animals and people are born with a set type of personality.
   
3. Environment - An enriched environment will help provide a positive experience and socialization.
   
4. In home work with crate training, potty training, leash and collar training, appropriate stuff to chew on, words, your household routine, and basic manners.
   
5. Training classes outside the home - Where you teach your dog obedience, commands, manners, and the ability to do everything with the distractions of other people and dog.
   
6. Serendipity - What the puppy learns on his own.